Focos Problem Library & Capability Atlas

Curated Technical Mind Training Bank

250 Curated Diverse Math Problems · 65 Capabilities · 419 Strategic Reasoning Checkpoints

Every problem features interactive stepwise discovery: propose mathematical ideas, explore strategic checkpoints, and verify governing invariants.
250 problems shown
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Difficulty:
algebracore
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Algebraic Manipulation · Algebra Problem 1

An equilateral triangle is inscribed in the ellipse x2+4y2=4x^2 + 4y^2 = 4 with one vertex at (0,1)(0,1) and the other two vertices on the ellipse having negative y-coordinates. The square of the length of the side of the triangle can be written as mn\frac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n.
#symmetry#coordinatevectors#transformation
ID: DIV-01
Solve Problem
algebracore
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Constrained Extremal Ratio in Quadratic System

Let aa and bb be positive real numbers with a≥ba\ge b. Let ρ\rho be the maximum possible value of ab\frac {a}{b} for which the system of equations a2+y2=b2+x2=(a−x)2+(b−y)2a^2 + y^2 = b^2 + x^2 = (a - x)^2 + (b - y)^2 has a solution in (x,y)(x,y) satisfying 0≤x<a0\le x < a and 0≤y<b0\le y < b. Then ρ2\rho^2 can be expressed as a fraction mn\frac {m}{n}, where mm and nn are [relatively prime](https://artofproblemsolving.com/wiki/index.php/Relatively_prime) positive integers. Find m+nm + n.
#bounding#coordinatevectors#objectto_graph
ID: DIV-02
Solve Problem
algebracore
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Algebraic Manipulation · Algebra Problem 3

Let z1=18+83i, z2=18+39i,z_1=18+83i,~z_2=18+39i, and z3=78+99i,z_3=78+99i, where i=−1.i=\sqrt{-1}. Let zz be the unique complex number with the properties that z3−z1z2−z1 ⋅ z−z2z−z3\frac{z_3-z_1}{z_2-z_1}~\cdot~\frac{z-z_2}{z-z_3} is a real number and the imaginary part of zz is the greatest possible. Find the real part of zz.
#modular_arithmetic#coordinatevectors#transformation
ID: DIV-03
Solve Problem
algebracore
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Case Analysis in Cyclic Geometric Configuration

What is the graph of y4+1=x4+2y2y^4+1=x^4+2y^2 in the coordinate plane? (A) two intersecting parabolas(B) two nonintersecting parabolas(C) two intersecting circles(D) a circle and a hyperbola(E) a circle and two parabolas\textbf{(A) } \text{two intersecting parabolas} \qquad \textbf{(B) } \text{two nonintersecting parabolas} \qquad \textbf{(C) } \text{two intersecting circles} \qquad \\\\ \textbf{(D) } \text{a circle and a hyperbola} \qquad \textbf{(E) } \text{a circle and two parabolas}
#case_analysis#coordinatevectors#parity
ID: DIV-04
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algebracore
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Algebraic Manipulation · Algebra Problem 5

For all integers xx and yy, let ax,ya_{x, y} be a real number. Suppose that a0,0=0a_{0, 0} = 0. Suppose that only a finite number of the ax,ya_{x, y} are nonzero. Prove that ∑x=−∞∞∑y=−∞∞ax,y(ax,2x+y+ax+2y,y)≤3∑x=−∞∞∑y=−∞∞ax,y2 . \sum_{x = -\infty}^\infty \sum_{y = -\infty}^{\infty} a_{x,y} ( a_{x, 2x + y} + a_{x + 2y, y} ) \le \sqrt{3} \sum_{x = -\infty}^\infty \sum_{y = -\infty}^{\infty} a_{x, y}^2 \, .
#bounding#contradiction#parity
ID: DIV-05
Solve Problem
algebracore
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Case Analysis in Cyclic Geometric Configuration

[b]p1.[/b] Let aa and bb be complex numbers such that a3+b3=−17a^3 + b^3 = -17 and a+b=1a + b = 1. What is the value of abab? [b]p2.[/b] Let AEFBAEFB be a right trapezoid, with ∠AEF=∠EAB=90o\angle AEF = \angle EAB = 90^o. The two diagonals EBEB and AFAF intersect at point DD, and CC is a point on AEAE such that AE⊥DCAE \perp DC. If AB=8AB = 8 and EF=17EF = 17, what is the lenght of CDCD? [b]p3.[/b] How many three-digit numbers abcabc (where each of aa, bb, and cc represents a single digit, a≠0a \ne 0) are there such that the six-digit number abcabcabcabc is divisible by 22, 33, 55, 77, 1111, or 1313? [b]p4.[/b] Let SS be the sum of all numbers of the form 1n\frac{1}{n} where nn is a postive integer and 1n\frac{1}{n} terminales in base bb, a positive integer. If SS is 158\frac{15}{8}, what is the smallest such bb? [b]p5.[/b] Sysyphus is having an birthday party and he has a square cake that is to be cut into 2525 square pieces. Zeus gets to make the first straight cut and messes up badly. What is the largest number of pieces Zeus can ruin (cut across)? Diagram? [b]p6.[/b] Given (9x2−y2)(9x2+6xy+y2)=16(9x^2 - y^2)(9x^2 + 6xy + y^2) = 16 and 3x+y=23x + y = 2. Find xyx^y. [b]p7.[/b] What is the prime factorization of the smallest integer NN such that N2\frac{N}{2} is a perfect square, N3\frac{N}{3} is a perfect cube, N5\frac{N}{5} is a perfect fifth power? [b]p8.[/b] What is the maximum number of pieces that an spherical watermelon can be divided into with four straight planar cuts? [b]p9.[/b] How many ordered triples of integers (x,y,z)(x,y,z) are there such that 0≤x,y,z≤1000 \le x, y, z \le 100 and (x−y)2+(y−z)2+(z−x)2≥(x+y−2z)+(y+z−2x)2+(z+x−2y)2.(x - y)^2 + (y - z)^2 + (z - x)^2 \ge (x + y - 2z) + (y + z - 2x)^2 + (z + x - 2y)^2. [b]p10.[/b] Find all real solutions to (2x−4)2+(4x−2)3=(4x+2x−6)3(2x - 4)^2 + (4x - 2)^3 = (4x + 2x - 6)^3. [b]p11.[/b] Let ff be a function that takes integers to integers that also has f(x)={x−5  if  x≥50f(f(x+12))  if  x<50f(x)=\begin{cases} x - 5\,\, if \,\, x \ge 50 \\ f (f (x + 12)) \,\, if \,\, x < 50 \end{cases} Evaluate f(2)+f(39)+f(58).f (2) + f (39) + f (58). [b]p12.[/b] If two real numbers are chosen at random (i.e. uniform distribution) from the interval [0,1][0,1], what is the probability that theit difference will be less than 35\frac35? [b]p13.[/b] Let aa, bb, and cc be positive integers, not all even, such that a<ba < b, b=c−2b = c - 2, and a2+b2=c2a^2 + b^2 = c^2. What is the smallest possible value for cc? [b]p14.[/b] Let ABCDABCD be a quadrilateral whose diagonals intersect at OO. If BO=8BO = 8, OD=8OD = 8, AO=16AO = 16, OC=4OC = 4, and AB=16AB = 16, then find ADAD. [b]p15.[/b] Let P0P_0 be a regular icosahedron with an edge length of 1717 units. For each nonnegative integer nn, recursively construct Pn+1P_{n+1} from Pn by performing the following procedure on each face of PnP_n: glue a regular tetrahedron to that face such that three of the vertices of the tetrahedron are the midpoints of the three adjacent edges of the face, and the last vertex extends outside of PnP_n. Express the number of square units in the surface area of P17P_{17} in the form uv⋅wxyz\frac{u^v\cdot w \sqrt{x}}{y^z} , where u,v,w,x,yu, v, w, x, y, and zz are integers, all greater than or equal to 22, that satisfy the following conditions: the only perfect square that evenly divides xx is 11, the GCD of uu and y is 11, and neither uu nor yy divides ww. Answers written in any other form will not be considered correct! PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
#case_analysis#factorization#coordinatevectors
ID: DIV-06
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algebracore
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Algebraic Manipulation · Algebra Problem 7

When nn standard six-sided dice are rolled, the product of the numbers rolled can be any of 936936 possible values. What is nn? (A) 11(B) 6(C) 8(D) 10(E) 9\textbf{(A)}~11\qquad\textbf{(B)}~6\qquad\textbf{(C)}~8\qquad\textbf{(D)}~10\qquad\textbf{(E)}~9
#symmetry#factorization#change_representation
ID: DIV-07
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algebraeasy
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Algebraic Manipulation · Algebra Problem 8

The graph of the even function f(x)f(x) is symmetric about the line x=2x=2, and f(3)=3f(3)=3. Find f(−1)f(-1).
#symmetry#parity#objectto_graph
ID: DIV-08
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algebraeasy
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Algebraic Manipulation · Algebra Problem 9

Given that f(x+2)f(x+2) is an even function, the axis of symmetry of the graph of y=f(2x)y=f(2x) is ( ) A: x=−1x=-1 B: x=1x=1 C: x=2x=2 D: x=−2x=-2
#symmetry#parity#objectto_graph
ID: DIV-09
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algebraeasy
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Algebraic Manipulation · Algebra Problem 10

Given that the function f(x)f(x) is an odd function on (−∞,+∞)(-\infty, +\infty), and the graph of f(x)f(x) is symmetric about x=1x=1. When x∈[0,1]x \in [0,1], f(x)=2x−1f(x) = 2^x - 1. Find the value of f(2012)+f(2013)f(2012) + f(2013). A: −2-2 B: −1-1 C: 00 D: 11
#symmetry#parity#objectto_graph
ID: DIV-10
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algebraeasy
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Algebraic Manipulation · Algebra Problem 11

Given the function f(x)=ln⁡x+ln⁡(2−x)f(x)=\ln x+\ln (2-x), which of the following statements is true? A: f(x)f(x) is monotonically increasing on the interval (0,2)(0,2) B: f(x)f(x) is monotonically decreasing on the interval (0,2)(0,2) C: The graph of y=f(x)y=f(x) is symmetric about the line x=1x=1 D: The graph of y=f(x)y=f(x) is symmetric about the point (1,0)(1,0)
#symmetry#objectto_graph
ID: DIV-11
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algebraeasy
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Algebraic Manipulation · Algebra Problem 12

Given a quadratic function y=ax2+bx+cy=ax^2+bx+c whose graph intersects the x-axis at points A(-2,0) and B(4,0), and the maximum value of the function is 9, find the expression of this quadratic function.
#symmetry#objectto_graph
ID: DIV-12
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algebraeasy
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Algebraic Manipulation · Algebra Problem 13

The graph of the function f(x)=x3−xf(x) = x^3 - x is symmetric about __________.
#symmetry#parity#objectto_graph
ID: DIV-13
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algebraeasy
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Case Analysis in Cyclic Geometric Configuration

Given the function f(x)=∣x−a∣f(x)=|x-a|, where a>1a > 1. (1) When a=3a=3, find the solution set of the inequality f(x)⩾4−∣x−4∣f(x)\geqslant 4-|x-4|. (2) If the area of the triangle formed by the graph of the function h(x)=f(2x+a)−2f(x)h(x)=f(2x+a)-2f(x) and the xx and yy axes is greater than a+4a+4, find the range of values for aa.
#case_analysis#bounding#objectto_graph
ID: DIV-14
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algebraeasy
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Algebraic Manipulation · Algebra Problem 15

If the function f(x)=x2−2(a+1)x+1f(x) = x^2 - 2(a+1)x + 1 is an even function, then a=a= ___.
#symmetry#parity#objectto_graph
ID: DIV-15
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algebraeasy
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Coordinates Vectors · Algebra Problem 16

If the graph of the function y=f(x)y=f(x) is translated by vector a\mathbf{a}, changing the coordinates of a point on the graph from (1,0)(1,0) to (2,2)(2,2), then the equation of the translated graph is ( ) A: y=f(x+1)−2y=f(x+1)-2 B: y=f(x−1)−2y=f(x-1)-2 C: y=f(x−1)+2y=f(x-1)+2 D: y=f(x+1)+2y=f(x+1)+2
#coordinatevectors#objectto_graph
ID: DIV-16
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algebraeasy
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Algebraic Manipulation · Algebra Problem 17

The graph of the function y=f(x)y=f(x) and its inverse function y=f−1(x)y=f^{-1}(x) are ( ) A: Symmetric about the y-axis B: Symmetric about the origin C: Symmetric about the line x+y=0x+y=0 D: Symmetric about the line x−y=0x-y=0
#symmetry#objectto_graph
ID: DIV-17
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algebraeasy
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Algebraic Manipulation · Algebra Problem 18

If the random variable ξ∼N(−1,σ2)\xi \sim N(-1, \sigma^2), and P(−3≤ξ≤−1)=0.4P(-3 \leq \xi \leq -1) = 0.4, then P(ξ≥1)= P(\xi \geq 1) = \ .
#symmetry#objectto_graph
ID: DIV-18
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algebraeasy
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Coordinates Vectors · Algebra Problem 19

The vertex of the parabola y=ax2+bx+cy = ax^2 + bx + c is (4,−11)(4, -11), and it intersects the x-axis at two points, one with a positive x-coordinate and the other with a negative x-coordinate. Among aa, bb, and cc, the positive number(s) is(are) (　　) A: only aa B: only bb C: only cc D: only aa and bb
#coordinatevectors#objectto_graph
ID: DIV-19
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algebraeasy
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Algebraic Manipulation · Algebra Problem 20

The monotonic decreasing interval of the function y=2−x2y=2-x^2 is ( ) A: (−∞，−1)(-\infty，-1) B: (−∞，0)(-\infty，0) C: (0，+∞)(0，+\infty) D: (−1，+∞)(-1，+\infty)
#symmetry#objectto_graph
ID: DIV-20
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algebraeasy
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Algebraic Manipulation · Algebra Problem 21

Among the following four options, the function f(x)=x2−1f(x) = x^2 - 1 is not monotonically decreasing on the interval: A: (−∞,−2)(-\infty, -2) B: (−2,−1)(-2, -1) C: (−1,1)(-1, 1) D: (−∞,0)(-\infty, 0)
#symmetry#parity#objectto_graph
ID: DIV-21
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algebraeasy
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Algebraic Manipulation · Algebra Problem 22

The graph of the function y=f(x)y = f(x) is symmetric to the graph of the function y=log⁡3xy = \log_{3}x (x>0x > 0) with respect to the line y=xy = x. Then, f(x)=________f(x) = \_\_\_\_\_\_\_\_.
#symmetry#objectto_graph
ID: DIV-22
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algebraeasy
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Case Analysis · Algebra Problem 23

Given the logarithmic function f(x)=log⁡ax(a>0f(x)=\log _{a}x (a > 0 and a≠1)a\neq 1), and its graph passes through the point (4,2)(4,2). (1) Find the value of the real number aa; (2) If f(x+1)<0f(x+1) < 0, find the range of the real number xx.
#case_analysis#objectto_graph
ID: DIV-23
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algebraeasy
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Coordinates Vectors · Algebra Problem 24

Given the parabola y=x2−(a+2)x+9y=x^2-(a+2)x+9 has its vertex on the x-axis, then a=a= ___.
#coordinatevectors#objectto_graph
ID: DIV-24
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algebraeasy
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Coordinates Vectors · Algebra Problem 25

In the Cartesian coordinate system, the graph of the quadratic function y=2x2y=2x^2 is translated upwards by 2 units. The equation of the resulting graph is (　　) A: y=2x2+2y=2x^2+2 B: y=2x2−2y=2x^2-2 C: y=2(x−2)2y=2(x-2)^2 D: y=2(x+2)2y=2(x+2)^2
#coordinatevectors#objectto_graph
ID: DIV-25
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algebraeasy
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Algebraic Manipulation · Algebra Problem 26

In the Cartesian coordinate system, the graph of the function y=∣x∣y=|x| is ( ) A: Not symmetric about either the coordinate axes or the origin B: Symmetric about the origin C: Symmetric about the x-axis D: Symmetric about the y-axis
#symmetry#coordinatevectors#parity
ID: DIV-26
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algebraeasy
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Objects To Graph · Algebra Problem 27

The domain of f(x)f(x) is [−2,3][-2, 3], and its range is [a,b][a, b]. Then, the range of y=f(x+4)y = f(x + 4) is ( ) A: [2,7][2, 7] B: [−6,−1][-6, -1] C: [a,b][a, b] D: [a+4,b+4][a + 4, b + 4]
#objectto_graph
ID: DIV-27
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algebraeasy
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Algebraic Manipulation · Algebra Problem 28

Given that the graph of the function y=g(x)y=g(x) is symmetric to the graph of the function y=3x+1y=3^x+1 about the line y=xy=x, find the value of g(10)g(10).
#symmetry#objectto_graph
ID: DIV-28
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algebraeasy
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Algebraic Manipulation · Algebra Problem 29

The quadratic function y=x2−2x+3y=x^2-2x+3 has the minimum ( ) value of ( ) A: Minimum 2 B: Maximum 2 C: Minimum 1 D: Maximum 1
#symmetry#objectto_graph
ID: DIV-29
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algebraeasy
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Algebraic Manipulation · Algebra Problem 30

Define a function f(x)f(x) on R\mathbb{R}, which is decreasing on (−∞,1)(-\infty, 1), and the graph of y=f(x)y=f(x) is symmetric about x=1x=1. The relationship between f(−1)f(-1) and f(3)f(3) is ( ). A: f(−1)>f(3)f(-1) > f(3) B: f(−1)<f(3)f(-1) < f(3) C: f(−1)=f(3)f(-1) = f(3) D: The relationship is uncertain
#symmetry#objectto_graph
ID: DIV-30
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algebraeasy
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Algebraic Manipulation · Algebra Problem 31

If the function y=f(x)y=f(x) is an odd function, then its graph must pass through the point ( ) A: (0,0)(0,0) B: (−a,−f(a))(-a,-f(a)) C: (a,f(−a))(a,f(-a)) D: (−a,−f(−a))(-a,-f(-a))
#parity#objectto_graph
ID: DIV-31
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algebraeasy
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Algebraic Manipulation · Algebra Problem 32

The monotonic increasing interval of the function f(x)=−x2+2x−2f(x) = -x^2 + 2x - 2 is __.
#symmetry#objectto_graph
ID: DIV-32
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algebraeasy
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Algebraic Manipulation · Algebra Problem 33

The graph of the function y=log⁡5xy=\log_{5}x and the graph of the function y=5xy=5^{x} are ( ) A: Symmetric about the origin B: Symmetric about the x-axis C: Symmetric about the y-axis D: Symmetric about the line y=xy=x
#symmetry#objectto_graph
ID: DIV-33
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algebraeasy
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Algebraic Manipulation · Algebra Problem 34

Given that the random variable XX follows a normal distribution N(2,σ2)N(2, \sigma^2), and P(X<4)=0.8P(X<4) = 0.8, then P(0<X<2)=________P(0<X<2) = \_\_\_\_\_\_\_\_.
#symmetry#objectto_graph
ID: DIV-34
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algebraeasy
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Algebraic Manipulation · Algebra Problem 35

The monotonic increasing interval of the function y=−x2y = -x^2 is ( ) A: (−∞,0](-\infty, 0] B: [0,+∞)[0, +\infty) C: (0,+∞)(0, +\infty) D: (−∞,+∞)(-\infty, +\infty)
#symmetry#objectto_graph
ID: DIV-35
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algebraeasy
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Algebraic Manipulation · Algebra Problem 36

Given y=f(x)y = f(x) is an increasing function on R\mathbb{R}, and A(0,−1)A(0, -1), B(3,1)B(3, 1) are two points on its graph, then the solution set of the inequality ∣f(x+1)∣<1|f(x + 1)| < 1 is __________.
#bounding#objectto_graph
ID: DIV-36
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algebraeasy
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Algebraic Manipulation · Algebra Problem 37

The solution set of the inequality x2+x−12≥0x^2 + x - 12 \geq 0 is (  ) A: {x∣x3}\{x|x3\} B: {x∣−4<x<3}\{x|-4<x<3\} C: {x∣x≤−4 or x≥3}\{x|x\leq-4 \text{ or } x\geq3 \} D: {x∣−4≤x≤3}\{x|-4\leq x\leq3\}
#bounding#objectto_graph
ID: DIV-37
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algebraeasy
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Algebraic Manipulation · Algebra Problem 38

The function f(x)f(x) satisfies f(x+5)=f(9−x)f(x+5)=f(9-x) for all real numbers xx. Therefore, the graph of f(x)f(x) is symmetric about __.
#symmetry#objectto_graph
ID: DIV-38
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algebraeasy
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Algebraic Manipulation · Algebra Problem 39

If the graph of the quadratic function y=ax2+bx+cy=ax^2+bx+c intersects the x-axis at A(−2,0)A(-2,0) and B(4,0)B(4,0), and the maximum value of the function is 99, then the expression of this quadratic function is __________.
#symmetry#objectto_graph
ID: DIV-39
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algebraeasy
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Algebraic Manipulation · Algebra Problem 40

The graph of the function f(x)f(x) is translated 1 unit to the right, and the resulting graph is symmetric about the y-axis with the curve y=exy = e^x. Then, f(x)=f(x) = (　　) A: ex+1e^{x+1} B: ex−1e^{x-1} C: e−x+1e^{-x+1} D: e−x−1e^{-x-1}
#symmetry#objectto_graph
ID: DIV-40
Solve Problem
algebraeasy
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Objects To Graph · Algebra Problem 41

For the function f(x)f(x), if f(−1)⋅f(3)<0f(-1) \cdot f(3) < 0, then ( ) A: The equation f(x)=0f(x)=0 definitely has a real solution. B: The equation f(x)=0f(x)=0 definitely has no real solution. C: The equation f(x)=0f(x)=0 definitely has two real roots. D: The equation f(x)=0f(x)=0 might have no real solution.
#objectto_graph#polynomial_roots
ID: DIV-41
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algebraeasy
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Algebraic Manipulation · Algebra Problem 42

Among the following functions, the graph that is symmetric to the graph of y=2xy=2^x with respect to the y-axis is ( ) A: y=−2xy=-2^x B: y=−2−xy=-2^{-x} C: y=2−xy=2^{-x} D: y=2x+2−xy=2^x+2^{-x}
#symmetry#objectto_graph
ID: DIV-42
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algebraeasy
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Coordinates Vectors · Algebra Problem 43

In a plane rectangular coordinate system, for any point (m, n) in the plane, the following two transformations are defined: ① f(m,n)=(m,−n)f(m, n)=(m, -n), for example, f(2,1)=(2,−1)f(2, 1)=(2, -1); ② g(m,n)=(−m,−n)g(m, n)=(-m, -n), for example, g(2,1)=(−2,−1)g(2, 1)=(-2, -1). According to these transformations, we have: f[g(3,4)]=f(−3,−4)=(−3,4)f[g(3, 4)]=f(-3, -4)=(-3, 4), then g[f(−3,2)]g[f(-3, 2)] equals to () A: (3,2)(3, 2) B: (3,−2)(3, -2) C: (−3,2)(-3, 2) D: (−3,−2)(-3, -2)
#coordinatevectors#transformation
ID: DIV-43
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algebraeasy
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Case Analysis · Algebra Problem 44

In the Cartesian coordinate system xOyxOy, there is a curve C1:x+y=4C_{1}: x+y=4, and another curve C2C_{2} defined by the parametric equations {x=1+cos⁡θ,y=sin⁡θ\begin{cases} x=1+\cos \theta, \\ y=\sin \theta \end{cases} (with θ\theta as the parameter). A polar coordinate system is established with the origin OO as the pole and the non-negative half-axis of xx as the polar axis. (1)(1) Find the polar equations of curves C1C_{1} and C2C_{2}. (2)(2) If a ray l:θ=α(ρ>0)l: \theta=\alpha (\rho > 0) intersects C1C_{1} and C2C_{2} at points AA and BB respectively, find the maximum value of ∣OB∣∣OA∣\dfrac{|OB|}{|OA|}.
#case_analysis#coordinatevectors
ID: DIV-44
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algebraeasy
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Algebraic Manipulation · Algebra Problem 45

Given the parabola y=−x2+bx+cy = -x^2 + bx + c has its axis of symmetry as the line x=3x = 3, then on the right side of the axis of symmetry, the function is _____. (Fill in "increasing" or "decreasing")
#symmetry#objectto_graph
ID: DIV-45
Solve Problem
algebraeasy
●●○○○

Algebraic Manipulation · Algebra Problem 46

Given the quadratic function y=−2(x+3)2+1y = -2(x+3)^2 + 1. Among the following statements: ① The graph opens downward; ② The axis of symmetry of the graph is the line x=3x = 3; ③ The coordinates of the vertex of the graph are (3,1)(3, 1); ④ As xx increases when x>3x > 3, yy decreases. The number of correct statements is (　　) A: 1 B: 2 C: 3 D: 4
#symmetry#objectto_graph
ID: DIV-46
Solve Problem
algebraeasy
●●○○○

Algebraic Manipulation · Algebra Problem 47

Given that the quadratic function f(x)f(x) has two zeros at 00 and −2-2, and the minimum value of f(x)f(x) is −1-1, the graph of function g(x)g(x) is symmetric to the graph of f(x)f(x) with respect to the origin. (1) Find the expressions for f(x)f(x) and g(x)g(x); (2) If h(x)=f(x)−λg(x)h(x) = f(x) - \lambda g(x) is an increasing function on the interval [−1,1][-1,1], find the range of the real number λ\lambda.
#symmetry#objectto_graph
ID: DIV-47
Solve Problem
algebraeasy
●●○○○

Algebraic Manipulation · Algebra Problem 48

In the same coordinate system, the graphs of the functions f(x)=31−xf(x) = 3^{1-x} and g(x)=31+xg(x) = 3^{1+x} are symmetric about ( ) A: the y-axis B: the line x=1x=1 C: the origin D: the x-axis
#symmetry#coordinatevectors#objectto_graph
ID: DIV-48
Solve Problem
algebraeasy
●●○○○

Algebraic Manipulation · Algebra Problem 49

In the same Cartesian coordinate system, it is known that the graph of the function y=f(x)y=f(x) is symmetric to the graph of y=exy=e^x with respect to the line y=xy=x. Then, the equation of the tangent line to the curve of the function y=f(x)y=f(x) at the point (e,f(e))(e, f(e)) is:
#symmetry#coordinatevectors#objectto_graph
ID: DIV-49
Solve Problem
algebraeasy
●●○○○

Algebraic Manipulation · Algebra Problem 50

The graph of y=f(x)y=f(x) is symmetric to the graph of the function g(x)=log⁡2xg(x)=\log_{2}x (x>0x>0) with respect to the origin. The expression for f(x)f(x) is.
#symmetry#objectto_graph
ID: DIV-50
Solve Problem
algebraeasy
●●○○○

Algebraic Manipulation · Algebra Problem 51

In the same coordinate system, the relationship between the graph of y=2xy=2^x and y=log⁡2xy=\log_2x is ( ) A: Symmetric about the y-axis B: Symmetric about the x-axis C: Symmetric about the origin D: Symmetric about the line y=xy=x
#symmetry#coordinatevectors#objectto_graph
ID: DIV-51
Solve Problem
algebraeasy
●●○○○

Algebraic Manipulation · Algebra Problem 52

Given the function y=x2−2x+8y=x^2-2x+8, then ( ) A: The function is monotonically increasing when x∈(1,+∞)x \in (1, +\infty) B: The function is monotonically decreasing when x∈(1,+∞)x \in (1, +\infty) C: The function is monotonically increasing when x∈(−∞,−1)x \in (-\infty, -1) D: The function is monotonically decreasing when x∈(−∞,3)x \in (-\infty, 3)
#symmetry#objectto_graph
ID: DIV-52
Solve Problem
algebraeasy
●●○○○

Algebraic Manipulation · Algebra Problem 53

Among the following propositions, ① Power functions are increasing functions in the first quadrant; ② The graph of a power function passes through the points (0, 0) and (1, 1); ③ If the power function y=xay=x^a is an odd function, then y=xay=x^a is an increasing function on its domain; ④ The graph of a power function cannot appear in the fourth quadrant. The correct proposition number is.
#parity#objectto_graph
ID: DIV-53
Solve Problem
algebrastretch
●●●○○

Algebraic Manipulation in Cyclic Geometric Configuration

23.11. (Finland, 80). Let on the coordinate plane be given a curve which is the graph of some polynomial P(x)=x4+px3+qx2+rx+s(p,q,r,s∈R) P(x)=x^{4}+p x^{3}+q x^{2}+r x+s(p, q, r, s \in \mathbf{R}) A line on this plane is called horizontal if it is parallel to the x-axis and intersects the curve at four points A,B,C,DA, B, C, D (listed from left to right). If, in addition, the lengths of the segments ABA B, ACA C, and ADA D can be the lengths of the sides of some triangle, then such a line is called triangular. Prove that only two cases are possible: either all horizontal lines are triangular, or none of them are triangular.
#symmetry#case_analysis#bounding
ID: DIV-54
Solve Problem
algebrastretch
●●●○○

Case Analysis · Algebra Problem 55

XV OM - III - Task 1 Prove that the inequality is not true for any value of α \alpha .
#case_analysis#bounding#coordinatevectors
ID: DIV-55
Solve Problem
combinatorics probabilitycore
●●●○○

Case Analysis · Combinatorics Problem 56

Four standard fair six-sided dice are rolled. The probability that the product of the four numbers rolled is a multiple of 1212 can be expressed as mn\frac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n.
#case_analysis#bounding#coordinatevectors
ID: DIV-56
Solve Problem
combinatorics probabilitycore
●●●○○

Counting Principles in Cyclic Geometric Configuration

A bug starts at a vertex of a grid made of equilateral triangles of side length 11. At each step the bug moves in one of the 66 possible directions along the grid lines randomly and independently with equal probability. What is the probability that after 55 moves the bug never will have been more than 11 unit away from the starting position? (A) 13108(B) 754(C) 29216(D) 427(E) 116\textbf{(A)}\ \frac{13}{108} \qquad\textbf{(B)}\ \frac{7}{54} \qquad\textbf{(C)}\ \frac{29}{216} \qquad\textbf{(D)}\ \frac{4}{27} \qquad\textbf{(E)}\ \frac{1}{16}
#symmetry#case_analysis#construction
ID: DIV-57
Solve Problem
combinatorics probabilitycore
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Counting Principles · Combinatorics Problem 58

There are n>1n>1 cities in the country, some pairs of cities linked two-way through straight flight. For every pair of cities there is exactly one aviaroute (can have interchanges). Major of every city X counted amount of such numberings of all cities from 11 to nn , such that on every aviaroute with the beginning in X, numbers of cities are in ascending order. Every major, except one, noticed that results of counting are multiple of 20162016. Prove, that result of last major is multiple of 20162016 too.
#contradiction#construction#gcd_structure
ID: DIV-58
Solve Problem
combinatorics probabilitycore
●●●○○

Case Analysis · Combinatorics Problem 59

Given any positive real number ε\varepsilon, prove that, for all but finitely many positive integers vv, any graph on vv vertices with at least (1+ε)v(1+\varepsilon)v edges has two distinct simple cycles of equal lengths. (Recall that the notion of a simple cycle does not allow repetition of vertices in a cycle.) [i]Fedor Petrov, Russia[/i]
#case_analysis#parity#construction
ID: DIV-59
Solve Problem
combinatorics probabilitycore
●●●○○

Case Analysis in Cyclic Geometric Configuration

A frog trainer places one frog at each vertex of an equilateral triangle ABCABC of unit sidelength. The trainer can make one frog jump over another along the line joining the two, so that the total length of the jump is an even multiple of the distance between the two frogs just before the jump. Let MM and NN be two points on the rays ABAB and ACAC, respectively, emanating from AA, such that AM=AN=ℓAM = AN = \ell, where ℓ\ell is a positive integer. After a fi nite number of jumps, the three frogs all lie in the triangle AMNAMN (inside or on the boundary), and no more jumps are performed. Determine the number of final positions the three frogs may reach in the triangle AMNAMN. (During the process, the frogs may leave the triangle AMNAMN, only their nal positions are to be in that triangle.)
#case_analysis#modular_arithmetic#coordinatevectors
ID: DIV-60
Solve Problem
combinatorics probabilitycore
●●●○○

Counting Principles · Combinatorics Problem 61

Let nn be a positive integer. Prove that (n0)−1+(n1)−1+⋯+(nn)−1=n+12n+1(21+222+⋯+2n+1n+1). \binom{n}{0}^{-1} + \binom{n}{1}^{-1} + \cdots + \binom{n}{n}^{-1} = \frac{n+1}{2^{n+1}} \left( \frac{2}{1} + \frac{2^2}{2} + \cdots + \frac{2^{n+1}}{n+1} \right).
#symmetry#case_analysis#coordinatevectors
ID: DIV-61
Solve Problem
combinatorics probabilitycore
●●●○○

Counting Principles · Combinatorics Problem 62

6. A square chessboard is composed of 2012×20122012 \times 2012 unit squares, and some of these squares are occupied by beetles, with at most one beetle per square. At a certain moment, all the beetles fly up and then land again on the squares of the chessboard, with at most one beetle per square. The vector from the center of the square where a beetle was before flying to the center of the square where it lands is called the "displacement vector" of the beetle, and the sum of all the beetles' displacement vectors is called the "total displacement vector." For all possible numbers of beetles and their initial and final positions, find the maximum length of the total displacement vector. （Zhai Zhenhua, Problem Supplier）
#symmetry#bounding#coordinatevectors
ID: DIV-62
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis · Combinatorics Problem 63

The number of distinct tetrahedra that can be formed using the vertices of a triangular prism is ___.
#case_analysis#objectto_graph
ID: DIV-63
Solve Problem
combinatorics probabilityeasy
●●○○○

Coordinates Vectors & Combinatorial Partitions

Two dice are rolled sequentially, with the scores obtained denoted as mm and nn respectively. Let θ\theta represent the angle between the vector a=(m,n)a=(m,n) and vector b=(1,−1)b=(1,-1). Calculate the probability that θ∈(0,π2]\theta \in (0, \frac{\pi}{2}].
#coordinatevectors#change_representation
ID: DIV-64
Solve Problem
combinatorics probabilityeasy
●●○○○

Counting Principles · Combinatorics Problem 65

We often organize and represent data collected from surveys using various types of statistical graphs. Among the following types of statistical graphs, the one that can highlight the relationship between a part and the whole represented by the data is: A: Bar graph B: Pie chart C: Line graph D: Frequency distribution histogram
#change_representation#objectto_graph
ID: DIV-65
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis · Combinatorics Problem 66

Given the sets M={1,−2,3}M=\{1, -2, 3\} and N={−4,5,6,−7}N=\{-4, 5, 6, -7\}, if one number is selected from each set to form a coordinate point, how many different points located in the third and fourth quadrants of the Cartesian coordinate system can be represented this way? A: 18 points B: 10 points C: 16 points D: 14 points
#case_analysis#coordinatevectors
ID: DIV-66
Solve Problem
combinatorics probabilityeasy
●●○○○

Counting Principles · Combinatorics Problem 67

To represent the percentage of a family's annual expenditures on "education," "clothing," "food," and "other" out of the total annual expenditures, choose the most suitable type of graph from a "pie chart," "bar graph," or "line graph."
#change_representation#objectto_graph
ID: DIV-67
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis & Combinatorial Partitions

Roll a die twice in succession, observing the upward facing number of points each time, and calculate: (I) The probability that the sum of the two numbers is 5; (II) If the upward facing number of points of the first roll is denoted as the x-coordinate, and of the second roll as the y-coordinate to form a point (x, y), find the probability that this point lies within the region Ω defined as: {x>0y>0x−y−2>0 \begin{cases} x>0 \\ y>0 \\ x-y-2>0 \end{cases}
#case_analysis#coordinatevectors
ID: DIV-68
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis & Combinatorial Partitions

Given a point M(a, b) in the Cartesian coordinate system xOy, where a is chosen from the numbers 1, 2, 3, and b is chosen from the numbers 1, 2, 3, 4. Define the event "point M(a, b) lies on the line x+y=n" as event QnQ_n (2≤n≤72 \leq n \leq 7, n is an integer). Then, when the probability of QnQ_n is the highest, all possible values of n are.
#case_analysis#coordinatevectors
ID: DIV-69
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis & Combinatorial Partitions

Xiaoli decides which subject among history, geography, or politics to review during tonight's self-study session based on the outcome of a mathematical game. The rules of the game are as follows: in the Cartesian coordinate system, starting from the origin OO, and then ending at points P1(−1,0)P_{1}(-1,0), P2(−1,1)P_{2}(-1,1), P3(0,1)P_{3}(0,1), P4(1,1)P_{4}(1,1), P5(1,0)P_{5}(1,0), to form 55 vectors. By randomly selecting any two vectors and calculating the dot product yy of these two vectors, if y>0y > 0, she will review history; if y=0y=0, she will review geography; if y<0y < 0, she will review politics. (1)(1) List all possible values of yy; (2)(2) Calculate the probability of Xiaoli reviewing history and the probability of reviewing geography.
#case_analysis#coordinatevectors
ID: DIV-70
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis in Cyclic Geometric Configuration

Color each vertex of a pyramid with a base that is a quadrilateral using one color, ensuring that the two vertices on the same edge are of different colors. Given that there are 5 different colors available, the total number of different coloring methods is ______. (Answer with a number)
#case_analysis#objectto_graph
ID: DIV-71
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis · Combinatorics Problem 72

In the two-dimensional Cartesian coordinate system, the coordinates of point P(a,b)P(a,b) satisfy a≠ba\neq b, and both aa and bb are elements of the set 1,2,3,4,5,6{1,2,3,4,5,6}. If the distance between point PP and the origin (∣OP∣⩾5|OP|\geqslant 5), then the number of such points PP is _______.
#case_analysis#coordinatevectors
ID: DIV-72
Solve Problem
combinatorics probabilityeasy
●●○○○

Counting Principles · Combinatorics Problem 73

We often organize and represent data collected from surveys using various types of statistical graphs. Among the following types of statistical graphs, the one that can highlight the relationship between a part and the whole represented by the data is (　　　): - A. Bar graph - B. Pie chart - C. Line graph - D. Frequency distribution histogram
#change_representation#objectto_graph
ID: DIV-73
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis & Combinatorial Partitions

Xiaoli decides which subject to review during her self-study session tonight—history, geography, or politics—based on the result of a mathematical game. The rules of the game are as follows: In the Cartesian coordinate plane, starting from the origin O and ending at the points P1 (−1, 0), P2 (−1, 1), P3 (0, 1), P4 (1, 1), P5 (1, 0), she forms 5 vectors. After choosing any two of these vectors, she calculates the dot product y of them. If y > 0, she will review history; if y = 0, she will review geography; and if y < 0, she will review politics. (1) List all possible values for y. (2) Calculate the probability of Xiaoli reviewing history and the probability of her reviewing geography.
#case_analysis#coordinatevectors
ID: DIV-74
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis · Combinatorics Problem 75

Given the variables xx and yy that satisfy the conditions {x+y−1<0  x−y⩽0  x⩾0\begin{cases} & x+y-1 < 0 \ \ & x-y\leqslant 0 \ \ & x\geqslant 0 \end{cases}, find the maximum value of 2x−y2x-y.
#case_analysis#objectto_graph
ID: DIV-75
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis & Combinatorial Partitions

The number of ways to select 4 non-coplanar points from the 10 points consisting of the vertices and the midpoints of the edges of a tetrahedron is ( ) A: 141 B: 144 C: 150 D: 210
#case_analysis#objectto_graph
ID: DIV-76
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis & Combinatorial Partitions

A chooses two vertices from the four vertices of a square to form a line, and B does the same with the square's four vertices. The probability that the two lines formed are perpendicular to each other is (    ) A: 13\dfrac{1}{3} B: 16\dfrac{1}{6} C: 536\dfrac{5}{36} D: 518\dfrac{5}{18}
#case_analysis#objectto_graph
ID: DIV-77
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis · Combinatorics Problem 78

How many four-digit even numbers can be formed using the digits 0, 1, 2 (all of them)?
#case_analysis#parity#similarity
ID: DIV-78
Solve Problem
combinatorics probabilityeasy
●●○○○

Counting Principles · Combinatorics Problem 79

Milk contains rich nutrients, with water accounting for approximately 82%82\%, protein approximately 4.3%4.3\%, fat approximately 6%6\%, lactose approximately 7%7\%, and other components approximately 0.7%0.7\%. These components play a very important role in human health. To visually represent the percentage of each component in the total, the most suitable type of graph is ______.
#change_representation#objectto_graph
ID: DIV-79
Solve Problem
combinatorics probabilityeasy
●●○○○

Counting Principles · Combinatorics Problem 80

Air is composed of a mixture of various gases. To visually represent the percentage of each component in the air, the most suitable type of graph to use is: A: Bar graph B: Line graph C: Pie chart D: Histogram
#change_representation#objectto_graph
ID: DIV-80
Solve Problem
combinatorics probabilityeasy
●●○○○

Coordinates Vectors · Combinatorics Problem 81

In the Cartesian coordinate system xOyxOy, the parallel lines x=nx=n (n=0,1,2,…,5n=0,1,2,\ldots,5) and the parallel lines y=ny=n (n=0,1,2,…,5n=0,1,2,\ldots,5) form a figure. The total number of rectangles in this figure is ( ) A: 25 B: 36 C: 100 D: 225
#coordinatevectors#objectto_graph
ID: DIV-81
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis in Cyclic Geometric Configuration

In △AOB\triangle AOB, there are 5 points on side OA, 6 points on side OB, plus point O, making a total of 12 points. The number of triangles that can be formed using these 12 points as vertices is     .
#case_analysis#objectto_graph
ID: DIV-82
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis in Cyclic Geometric Configuration

The center and vertices of a regular hexagon make up 7 points in total. The number of triangles that can be formed using any 3 of these points is ( ). A: 35 B: 32 C: 210 D: 207
#case_analysis#objectto_graph
ID: DIV-83
Solve Problem
combinatorics probabilityeasy
●●○○○

Counting Principles · Combinatorics Problem 84

Find the value of the combinatorial expression Cn0−2Cn1+4Cn2−8Cn3+…+(−2)nCnn.C_n^0 - 2C_n^1 + 4C_n^2 - 8C_n^3 + \ldots + (-2)^n C_n^n. A: (−1)n(-1)^n B: 1 C: 3n3^n D: 3n−13^n - 1
#factorization#parity#transformation
ID: DIV-84
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis & Combinatorial Partitions

Two dice are rolled consecutively, and the scores obtained are represented by mm and nn respectively. If the angle between the vector a→=(m,n)\overrightarrow{a} = (m, n) and the vector b→=(1,−2)\overrightarrow{b} = (1, -2) is represented by θ\theta, find the probability that θ\theta is an acute angle.
#case_analysis#coordinatevectors
ID: DIV-85
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis · Combinatorics Problem 86

From the eight vertices of the cube ABCD−A1B1C1D1ABCD-A_1B_1C_1D_1, select four as the vertices of a tetrahedron. The number of different tetrahedrons that can be formed is __________. (Answer with an expression)
#case_analysis#objectto_graph
ID: DIV-86
Solve Problem
combinatorics probabilityeasy
●●○○○

Counting Principles · Combinatorics Problem 87

The variation of H fractals follows a certain rule: each "H" will transform into two additional "Hs". Count the number of "H" in each H fractal figure to form a sequence {ana_n}. Given that the first three terms of the sequence {ana_n} are a1=1a_1=1, a2=3a_2=3, and a3=7a_3=7, determine the value of a4a_4 ( ). A: 11 B: 15 C: 19 D: 23
#transformation#change_representation#induction
ID: DIV-87
Solve Problem
combinatorics probabilityeasy
●●○○○

Counting Principles · Combinatorics Problem 88

According to statistics, the global forest area has been decreasing at a rate of approximately 17 million hectares per year in recent years. To predict the trend of changes in the world's forest area over the next 20 years, which type of statistical graph can be used to represent the collected data?
#change_representation#objectto_graph
ID: DIV-88
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis · Combinatorics Problem 89

If the vertices of a pyramid with a square base are colored, and two vertices connected by an edge are called adjacent vertices, with the rule that adjacent vertices cannot be colored the same color, and there are 4 colors to choose from, then the total number of different coloring methods is ( ) A: 36 B: 54 C: 72 D: 120
#case_analysis#objectto_graph
ID: DIV-89
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis · Combinatorics Problem 90

Given the sets M={1,−2,3}M=\{1,-2,3\} and N={−4,5,6,−7}N=\{-4,5,6,-7\}, if one number is selected from each set to form a coordinate point, the number of different points that can be represented in the third and fourth quadrants of the Cartesian coordinate system is (    ) A: 1818 points B: 1010 points C: 1616 points D: 1414 points
#case_analysis#coordinatevectors
ID: DIV-90
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis & Combinatorial Partitions

A regular tetrahedron has 1010 points consisting of its vertices and the midpoints of its edges. The number of ways to choose 44 non-coplanar points from these 1010 points is ______\_\_\_\_\_\_.
#case_analysis#objectto_graph
ID: DIV-91
Solve Problem
combinatorics probabilityeasy
●●○○○

Coordinates Vectors · Combinatorics Problem 92

In the XOY plane of the Cartesian coordinate system, the parallel lines (x=n)(x=n), (n=0,1,2…5)(n=0,1,2…5) and the parallel lines (y=n)(y=n), (n=0,1,2…5)(n=0,1,2…5) form rectangles. There are (____) rectangles in total. A: 100100 B: 225225 C: 150150 D: 420420
#coordinatevectors#similarity
ID: DIV-92
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis · Combinatorics Problem 93

Among the vertices of a cube, the number of different tetrahedrons that can be formed is (　　) A: C84C_{ 8 }^{ 4 } B: C81C73C_{ 8 }^{ 1 } C_{ 7 }^{ 3 } C: C84−6C_{ 8 }^{ 4 }-6 D: C84−12C_{ 8 }^{ 4 }-12
#case_analysis#objectto_graph
ID: DIV-93
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis & Combinatorial Partitions

The vertices and the midpoints of the edges of a tetrahedron make up 10 points in total. If we choose 4 non-coplanar points from these, the total number of different ways to do this is ( ) A: 150 ways B: 147 ways C: 144 ways D: 141 ways
#case_analysis#objectto_graph
ID: DIV-94
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis in Cyclic Geometric Configuration

In △AOB, there are 4 points on edge OA and 5 points on edge OB, including point O, making a total of 10 points. The number of triangles formed by taking these 10 points as vertices is ______\_\_\_\_\_\_.
#case_analysis#objectto_graph
ID: DIV-95
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis · Combinatorics Problem 96

How many tetrahedrons can be formed using the vertices of a regular triangular prism?
#case_analysis#objectto_graph
ID: DIV-96
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis · Combinatorics Problem 97

How many tetrahedrons can be formed by choosing four vertices from the six vertices of a triangular prism? A: 18 B: 15 C: 12 D: 9
#case_analysis#objectto_graph
ID: DIV-97
Solve Problem
combinatorics probabilityeasy
●●○○○

Coordinates Vectors in Cyclic Geometric Configuration

Given a∈{3,4,6}a \in \{3, 4, 6\}, b∈{1,2,7,8}b \in \{1, 2, 7, 8\}, r∈{5,9}r \in \{5, 9\}, then the equation (x−a)2+(y−b)2=r2(x-a)^2+(y-b)^2=r^2 can represent ( ) different circles. A: 36 B: 24 C: 12 D: 6
#coordinatevectors#change_representation
ID: DIV-98
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis · Combinatorics Problem 99

Given a pyramid with a vertex and base ABCD, each vertex is painted with one color, ensuring that two vertices on the same edge are of different colors. There are 5 different colors available. Calculate the total number of distinct coloring methods. (Answer with a number)
#case_analysis#objectto_graph
ID: DIV-99
Solve Problem
combinatorics probabilityeasy
●●○○○

Counting Principles · Combinatorics Problem 100

Air is a mixture. To visually represent the percentage of each component of air, the most suitable type of statistical graph is: A: Line graph B: Bar graph C: Histogram D: Pie chart
#change_representation#objectto_graph
ID: DIV-100
Solve Problem
combinatorics probabilityeasy
●●○○○

Coordinates Vectors · Combinatorics Problem 101

Given sets M={1,2,3}M=\{1,2,3\} and N={1,5}N=\{1,5\}, if one element is taken from each of these two sets to form the coordinates of a point, then the number of different points that can be determined in the Cartesian coordinate system is ( ) A: 11 B: 12 C: 6 D: 5
#coordinatevectors#change_representation
ID: DIV-101
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis in Cyclic Geometric Configuration

A regular hexagon's center and vertices together make 7 points. The number of triangles that can be formed using any 3 of these points as vertices is ( ). A: 35 B: 32 C: 210 D: 207
#case_analysis#objectto_graph
ID: DIV-102
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis & Combinatorial Partitions

A random simulation method is used to estimate the probability of an athlete hitting a target in a shooting event. A calculator generates random integers between 0 and 9. The numbers 0, 1, 2, and 3 represent a miss, while 4, 5, 6, 7, 8, and 9 represent a hit. Each group of 4 random numbers corresponds to the results of 4 shots. After random simulation, 20 groups of the following random numbers are generated: 7527, 0293, 7140, 9857, 0347, 4373, 8636, 6947, 1417, 4698, 0371, 6233, 2616, 8045, 6011, 3661, 9597, 7424, 7610, 4281 Estimate the probability that the athlete hits the target at least three times in four shots: ______\_\_\_\_\_\_.
#case_analysis#transformation#change_representation
ID: DIV-103
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis · Combinatorics Problem 104

A triangular prism ABC-DEF is to be painted with five different colors (not necessarily using all colors). Each vertex must be painted with one color, and the two endpoints of each edge must be painted with different colors. Determine the number of different painting schemes. A: 840 B: 1200 C: 1800 D: 1920
#case_analysis#objectto_graph
ID: DIV-104
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis & Combinatorial Partitions

Given a regular tetrahedron ABCDABCD with 66 edges, where the midpoints of the edges and the four vertices form a total of 1010 points. If any 44 points are selected from these 1010 points, find the probability that these four points are not coplanar. A: 57\frac{5}{7} B: 710\frac{7}{10} C: 2435\frac{24}{35} D: 4770\frac{47}{70}
#case_analysis#objectto_graph
ID: DIV-105
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis · Combinatorics Problem 106

How many tetrahedrons can be formed using the vertices of a regular triangular prism? A: 6 B: 12 C: 18 D: 30
#case_analysis#objectto_graph
ID: DIV-106
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis & Combinatorial Partitions

Given the function y=x−5y = x - 5, let x=0.5,1,1.5,2,2.5,3,3.5,4,4.5,5x = 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5, we can obtain 10 points on the graph of the function. Randomly select two points P(a,b)P(a, b) and Q(m,n)Q(m, n) from these 10 points. What is the probability that PP and QQ lie on the same inverse proportion function graph?
#case_analysis#objectto_graph
ID: DIV-107
Solve Problem
combinatorics probabilityeasy
●●○○○

Counting Principles · Combinatorics Problem 108

Given that the random variable ξ\xi follows a normal distribution N(0,σ2)N(0,\sigma^{2}), and P(−2⩽ξ⩽2)=0.4P(-2 \leqslant \xi \leqslant 2)=0.4, then P(ξ>2)=P(\xi > 2)= \_\_\_\_\_\_.
#symmetry#objectto_graph
ID: DIV-108
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis · Combinatorics Problem 109

In the spatial rectangular coordinate system O-xyz, there are 8 points: P1(1, 1, 1), P2(-1, 1, 1), …, P7(-1, -1, -1), P8(1, -1, -1) (where the x, y, and z coordinates of each point are either 1 or -1). Count the number of tetrahedra that can be formed with any four of these points as vertices.
#case_analysis#coordinatevectors#objectto_graph
ID: DIV-109
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis · Combinatorics Problem 110

In the same Cartesian coordinate system, there are infinitely many lines described by the equation y=kx+by = kx + b (k≠0k \neq 0, where kk and bb are real numbers). No matter how these lines are selected, the question is: how many lines must be taken at least to ensure that there are two lines passing through exactly the same quadrants? A: 4 lines B: 5 lines C: 6 lines D: 7 lines
#case_analysis#coordinatevectors
ID: DIV-110
Solve Problem
combinatorics probabilityeasy
●●○○○

Case Analysis & Combinatorial Partitions

In an experiment of throwing 2 dice, let the coordinate of point P be represented by (x,y)(x, y), where xx is the number shown on the first die, and yy is the number shown on the second die. - (I) Find the probability that point P lies on the line y=xy = x. - (II) Find the probability that point P satisfies x+y≥10x + y \geq 10.
#case_analysis#coordinatevectors
ID: DIV-111
Solve Problem
combinatorics probabilityeasy
●●○○○

Counting Principles · Combinatorics Problem 112

China's terrain is higher in the west and lower in the east. To visually represent the changes in elevation of various locations on the 32∘32^{\circ} north latitude line, the most suitable type of graph is ______ graph (fill in "bar", "pie", or "line").
#change_representation#objectto_graph
ID: DIV-112
Solve Problem
combinatorics probabilityshowcase
●●●●○

Case Analysis · Combinatorics Problem 113

The rows and columns of a 2n×2n2^{n} \times 2^{n} table are numbered from 0 to 2n−12^{n}-1. The cells of the table have been colored with the following property being satisfied: for each 0≤i,j≤2n−10 \leq i, j \leq 2^{n}-1, the jj th cell in the ii th row and the jj th cell in the ii th column have the same color. Prove that the maximal possible number of colors is 2n2^{n}.
#case_analysis#factorization#modular_arithmetic
ID: DIV-113
Solve Problem
combinatorics probabilitystretch
●●●○○

Coordinates Vectors · Combinatorics Problem 114

22 * On an infinite sheet of white squared paper, there are a finite number of squares colored black, and each black square has an even number (0, 2, or 1) of white squares sharing a common edge with it. Prove: It is possible to color the remaining white squares red or green, such that for each black square, the number of red and green neighboring squares are equal (squares sharing a common edge are called adjacent).
#coordinatevectors#contradiction#parity
ID: DIV-114
Solve Problem
combinatorics probabilitystretch
●●●○○

Counting Principles · Combinatorics Problem 115

Exercise 5. Consider a 4×44 \times 4 square grid. Two distinct cells that share a common side are called neighbors. Initially, all cells are red. Subsequently, a cell can potentially change color and be either red or blue. Performing an operation on cell cc means simultaneously changing the color of cc and all its neighbors (red cells become blue and blue cells become red). Give all values of n\mathrm{n} for which there exists a group of n\mathrm{n} distinct cells such that after performing an operation on each of these nn cells, one obtains a grid that is entirely blue.
#symmetry#modular_arithmetic#contradiction
ID: DIV-115
Solve Problem
geometrycore
●●●○○

Similarity Strategy · Geometry Problem 116

Define a regular nn-pointed star to be the union of nn line segments P1P2,P2P3,…,PnP1P_1P_2, P_2P_3,\ldots, P_nP_1 such that the points P1,P2,…,PnP_1, P_2,\ldots, P_n are coplanar and no three of them are collinear, each of the nn line segments intersects at least one of the other line segments at a point other than an endpoint, all of the angles at P1,P2,…,PnP_1, P_2,\ldots, P_n are congruent, all of the nn line segments P1P2,P2P3,…,PnP1P_1P_2, P_2P_3,\ldots, P_nP_1 are congruent, and the path P1P2,P2P3,…,PnP1P_1P_2, P_2P_3,\ldots, P_nP_1 turns counterclockwise at an angle of less than 180 degrees at each vertex. There are no regular 3-pointed, 4-pointed, or 6-pointed stars. All regular 5-pointed stars are similar, but there are two non-similar regular 7-pointed stars. How many non-similar regular 1000-pointed stars are there?
#symmetry#case_analysis#modular_arithmetic
ID: DIV-116
Solve Problem
geometrycore
●●●○○

Similarity Strategy · Geometry Problem 117

In a geometric configuration with perpendicular chords and tangent segments of lengths 1212 and 1313, compute the integer invariant sum m+nm+n.
#symmetry#modular_arithmetic#coordinatevectors
ID: DIV-117
Solve Problem
geometrycore
●●●○○

Similarity Strategy · Geometry Problem 118

Rodrigo has a very large sheet of graph paper. First he draws a line segment connecting point (0,4)(0,4) to point (2,0)(2,0) and colors the 44 cells whose interiors intersect the segment, as shown below. Next Rodrigo draws a line segment connecting point (2000,3000)(2000,3000) to point (5000,8000)(5000,8000). How many cells will he color this time? (A) 6000(B) 6500(C) 7000(D) 7500(E) 8000\textbf{(A) } 6000\qquad\textbf{(B) } 6500\qquad\textbf{(C) } 7000\qquad\textbf{(D) } 7500\qquad\textbf{(E) } 8000
#modular_arithmetic#coordinatevectors#contradiction
ID: DIV-118
Solve Problem
geometrycore
●●●○○

Case Analysis · Geometry Problem 119

For positive integers m,n≥2m,n \geq 2, let Sm,n={(i,j):i∈{1,2,…,m},j∈{1,2,…,n}}S_{m,n} = \{(i,j): i \in \{1,2,\ldots,m\}, j\in \{1,2,\ldots,n\}\} be a grid of mnmn lattice points on the coordinate plane. Determine all pairs (m,n)(m,n) for which there exists a simple polygon PP with vertices in Sm,nS_{m,n} such that all points in Sm,nS_{m,n} are on the boundary of PP, all interior angles of PP are either 90∘90^{\circ} or 270∘270^{\circ} and all side lengths of PP are 11 or 33.
#case_analysis#modular_arithmetic#coordinatevectors
ID: DIV-119
Solve Problem
geometrycore
●●●○○

Case Analysis in Cyclic Geometric Configuration

In an isosceles right-angled triangle shaped billiards table , a ball starts moving from one of the vertices adjacent to hypotenuse. When it reaches to one side then it will reflect its path. Prove that if we reach to a vertex then it is not the vertex at initial position [i]By Sam Nariman[/i]
#case_analysis#modular_arithmetic#coordinatevectors
ID: DIV-120
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 121

The coordinates of the point symmetric to point P(2, -3, -5) with respect to the y-axis are.
#symmetry#coordinatevectors
ID: DIV-121
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 122

Given point A (-4, 8, 6), the coordinates of the point symmetric to A with respect to the y-axis are (    ) A: (4, 8, -6) B: (-4, -8, -6) C: (-6, -8, 4) D: (-4, -8, 6)
#symmetry#coordinatevectors
ID: DIV-122
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 123

In the space rectangular coordinate system, the coordinates of the point P(1, 2, 3) symmetric to the plane xoz are ( ) A: (-1, 2, 3) B: (-1, -2, 3) C: (1, -2, 3) D: (1, 2, -3)
#symmetry#coordinatevectors
ID: DIV-123
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 124

In the space rectangular coordinate system O-xyz, the coordinates of point M' which is symmetric to point M(-1, 2, 1) with respect to the x-axis are ( ) A: (-1, -2, -1) B: (1, -2, 1) C: (-1, 2, -1) D: (1, -2, -1)
#symmetry#coordinatevectors
ID: DIV-124
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 125

In the space rectangular coordinate system Oxyz, the coordinates of the point symmetric to point P(3, 1, 5) with respect to the yOz plane are.
#symmetry#coordinatevectors
ID: DIV-125
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 126

In the three-dimensional Cartesian coordinate system, what are the coordinates of the point symmetric to the point (3, -4, 1) with respect to the y-axis?
#symmetry#coordinatevectors
ID: DIV-126
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 127

In the Cartesian coordinate system, point A is given as (3, 2). The symmetric point of A about the y-axis is denoted as A1, the symmetric point of A1 about the origin is denoted as A2, the symmetric point of A2 about the x-axis is denoted as A3, and the symmetric point of A3 about the y-axis is denoted as A4, and so on following this pattern. Then, the coordinates of point A8 are.
#symmetry#coordinatevectors
ID: DIV-127
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 128

In the spatial Cartesian coordinate system, the coordinates of the point symmetric to point P(1, 2, 3) with respect to the x-axis are ( ) A: (1, 2, 3) B: (-1, -2, -3) C: (1, -2, -3) D: (-1, 2, 3)
#symmetry#coordinatevectors
ID: DIV-128
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 129

In the three-dimensional Cartesian coordinate system, if P(3,−2,1)P(3,-2,1), then the coordinates of the point symmetric to PP with respect to the xOzxOz plane are: (     ) A: (−3,−2,−1)(-3,-2,-1) B: (3,2,1)(3,2,1) C: (−3,2,−1)(-3,2,-1) D: (3,−2,−1)(3,-2,-1)
#symmetry#coordinatevectors
ID: DIV-129
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 130

In the Cartesian coordinate system, point M (3, -4) is symmetric to point N with respect to the y-axis. The coordinates of point N are (　　) A: (3, 4) B: (-3, -4) C: (-3, 4) D: (4, -3)
#symmetry#coordinatevectors
ID: DIV-130
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 131

In the three-dimensional Cartesian coordinate system, the coordinates of the point symmetric to point A(1, 1, 2) with respect to the x-axis are ____.
#symmetry#coordinatevectors
ID: DIV-131
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 132

In the Cartesian coordinate system, the coordinates of the point symmetric to point P(2, 3) with respect to the origin are.
#symmetry#coordinatevectors
ID: DIV-132
Solve Problem
geometryeasy
●●○○○

Similarity Strategy in Cyclic Geometric Configuration

Given that the coordinates of points A and B are (2, 0) and (2, 4) respectively, write down the coordinates of a point P such that the triangle formed by vertices A, B, and P is congruent to triangle △ABO:
#modular_arithmetic#objectto_graph
ID: DIV-133
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 134

In the three-dimensional Cartesian coordinate system, the coordinates of the point symmetric to point P(1, 3, -5) with respect to the origin O are (    ) A: (-1, 3, -5) B: (1, -3, 5) C: (1, 3, 5) D: (-1, -3, 5)
#symmetry#coordinatevectors
ID: DIV-134
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 135

In the three-dimensional Cartesian coordinate system, the positional relationship between point P (2, 3, 4) and point Q (2, -3, 4) is ( ) A: Symmetric about the x-axis B: Symmetric about the xOy plane C: Symmetric about the coordinate origin D: Symmetric about the xOz plane
#symmetry#coordinatevectors
ID: DIV-135
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 136

In the three-dimensional Cartesian coordinate system, the coordinates of point M are (4,7,6)(4, 7, 6). What are the coordinates of the projection of the point symmetric to M with respect to the y-axis on the xOz coordinate plane? A: (4,0,6)(4, 0, 6) B: (−4,7,−6)(-4, 7, -6) C: (−4,0,−6)(-4, 0, -6) D: (−4,7,0)(-4, 7, 0)
#symmetry#coordinatevectors
ID: DIV-136
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 137

It is known that point P has coordinates (2, 3), and the symmetric point of P with respect to the y-axis is ____.
#symmetry#coordinatevectors
ID: DIV-137
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 138

The graph of the function y=sin⁡xy = \sin x is ( ) A: Symmetric about the line x=−x = - B: Symmetric about the line x=−x = - C: Symmetric about the line x=x = D: Symmetric about the line x=πx = \pi
#symmetry#objectto_graph
ID: DIV-138
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 139

In the three-dimensional Cartesian coordinate system O-xyz, suppose point M is the symmetric point of point N (2, -3, 5) with respect to the coordinate plane xoy, then the length of the line segment MN is     .
#symmetry#coordinatevectors
ID: DIV-139
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 140

In the Cartesian coordinate system, the equation of the line symmetric to the line y=2x+1y = 2x + 1 about the point (1,1)(1,1) is (　　) A: y=2x−1y = 2x - 1 B: y=−2x+1y = -2x + 1 C: y=−2x+3y = -2x + 3 D: y=2x−3y = 2x - 3
#symmetry#coordinatevectors
ID: DIV-140
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 141

In the three-dimensional Cartesian coordinate system, the coordinates of the point symmetric to point P(3, 1, 5) with respect to the yOz plane are (   ) A: (﹣3, 1, 5) B: (﹣3, ﹣1, 5) C: (3, ﹣1, ﹣5) D: (﹣3, 1, ﹣5)
#symmetry#coordinatevectors
ID: DIV-141
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 142

In the Cartesian coordinate system, the distance between the point (2, -3) and its symmetric point about the x-axis is.
#symmetry#coordinatevectors
ID: DIV-142
Solve Problem
geometryeasy
●●○○○

Case Analysis in Cyclic Geometric Configuration

In the Cartesian coordinate system xoyxoy, the parametric equation of line ll is {x=1+tcos⁡θy=tsin⁡θ\begin{cases} x=1+t\cos\theta \\ y=t\sin\theta\end{cases} (where tt is the parameter, 0≤θ<π0\leq\theta<\pi). Taking the origin as the pole and the non-negative half-axis of xx as the polar axis, we establish a polar coordinate system. The polar equation of circle CC is ρ=−4cos⁡α\rho=-4\cos\alpha. The distance from the center of circle CC to line ll is 32\frac{3}{2}. (1) Find the value of θ\theta; (2) Given P(1,0)P(1,0), if line ll intersects circle CC at points AA and BB, find the value of 1∣PA∣+1∣PB∣\frac{1}{|PA|}+ \frac{1}{|PB|}.
#case_analysis#coordinatevectors
ID: DIV-143
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 144

Given the point M(-2, -3), the symmetric point of M with respect to the x-axis is ( ) A: (2, -3) B: (-2, 3) C: (-2, -3) D: (2, 3)
#symmetry#coordinatevectors
ID: DIV-144
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 145

In the three-dimensional Cartesian coordinate system, the coordinates of the point symmetric to point A(1,2,1) with respect to the x-axis are A: (-1,2,1) B: (-1,-2,1) C: (1,-2,-1) D: (1,2,-1)
#symmetry#coordinatevectors
ID: DIV-145
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 146

In the Cartesian coordinate system, the coordinates of the point symmetrical to point P (-1, 2) with respect to the origin are ( ) A: (-1, -2) B: (1, 2) C: (2, -1) D: (1, -2)
#symmetry#coordinatevectors
ID: DIV-146
Solve Problem
geometryeasy
●●○○○

Coordinates Vectors in Cyclic Geometric Configuration

In the Cartesian coordinate system, point A has coordinates (4, 0), and point P is on the graph of the linear function y=−x+6y = -x + 6 in the first quadrant. The origin is O. If the area of △OPA\triangle OPA is SS, and the coordinates of point P are (x,y)(x, y), find the expression of SS as a function of xx.
#coordinatevectors#objectto_graph
ID: DIV-147
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 148

In the three-dimensional Cartesian coordinate system, point A (2, 3, 1) is symmetric to point B with respect to the yOz coordinate plane. Find the length of ∣AB∣|AB|.
#symmetry#coordinatevectors
ID: DIV-148
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 149

In the Cartesian coordinate system, given that point A (m,3)(m, 3) and point B (4,n)(4, n) are symmetric about the y-axis, the value of (m+n)2015(m+n)^{2015} is (　　) A: −1-1 B: 11 C: −72015-7^{2015} D: 720157^{2015}
#symmetry#coordinatevectors
ID: DIV-149
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 150

Given a point M(a,b,c)M(a,b,c) in the three-dimensional Cartesian coordinate system O−xyzO-xyz, the coordinates of the point symmetric to MM with respect to the zz axis are (    ) A: (a,−b,−c)(a,-b,-c) B: (−a,b,−c)( -a,b,-c) C: (−a,−b,c)( -a,-b,c) D: (−a,−b,−c)( -a,-b,-c)
#symmetry#coordinatevectors
ID: DIV-150
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 151

Given that the coordinates of point P1P_1, which is symmetric to point PP with respect to the origin, are (−2,3)(-2, 3), then the coordinates of point P2P_2, which is symmetric to point PP with respect to the y-axis, are (　　) A: (−3,−2)(-3, -2) B: (2,−3)(2, -3) C: (−2,−3)(-2, -3) D: (−2,3)(-2, 3)
#symmetry#coordinatevectors
ID: DIV-151
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 152

The coordinates of the point P(2, 5) symmetric to the _x_-axis are (     ) A: (5, 2) B: (－2, 5) C: (2, －5) D: (－5, －2)
#symmetry#coordinatevectors
ID: DIV-152
Solve Problem
geometryeasy
●●○○○

Similarity Strategy in Cyclic Geometric Configuration

Among the negations of the following statements, the incorrect one is ( ) A: P: An integer divisible by 3 is odd; ┐P: There exists an integer divisible by 3 that is not odd. B: P: There exists a quadrilateral whose four vertices do not lie on a common circle; ┐P: Every quadrilateral's four vertices lie on a common circle. C: P: Some triangles are equilateral; ┐P: Not all triangles are equilateral. D: P: ∃x∈R,x2+2x+2≤0\exists x \in \mathbb{R}, x^2+2x+2 \leq 0; ┐P: ∀x∈R,x2+2x+2>0\forall x \in \mathbb{R}, x^2+2x+2 > 0
#parity#objectto_graph
ID: DIV-153
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 154

In the Cartesian coordinate system, it is known that point A (m,n,1)(m,n,1) and point B (3,2,1)(3,2,1) are symmetric with respect to the zz-axis. Find the value of m+nm+n.
#symmetry#coordinatevectors
ID: DIV-154
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 155

In the three-dimensional Cartesian coordinate system, if P(3,−2,1)P(3,-2,1), then the coordinates of the point symmetric to PP with respect to the xOzxOz plane are: A: (−3,−2,−1)(-3,-2,-1) B: (3,2,1)(3,2,1) C: (−3,2,−1)(-3,2,-1) D: (3,−2,−1)(3,-2,-1)
#symmetry#coordinatevectors
ID: DIV-155
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 156

If the number of faces of an n-sided prism is 10, then n=    , this prism has    vertices, and a total of    edges.
#modular_arithmetic#objectto_graph
ID: DIV-156
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 157

Given a point M(2, 1, 3) in the Cartesian coordinate system O-xyz, the coordinates of the point symmetric to M with respect to the x-axis are (   ) A: (2, -1, -3) B: (-2, 1, -3) C: (-2, -1, 3) D: (-2, -1, -3)
#symmetry#coordinatevectors
ID: DIV-157
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 158

The symmetric point of (1, 1, 1) with respect to the z-axis is () A: (-1, -1, 1) B: (1, -1, -1) C: (-1, 1, -1) D: (-1, -1, -1)
#symmetry#coordinatevectors
ID: DIV-158
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 159

In the spatial rectangular coordinate system, what are the coordinates of the point symmetric to point A(2, -3, 4) with respect to the yOz plane?
#symmetry#coordinatevectors
ID: DIV-159
Solve Problem
geometryeasy
●●○○○

Coordinates Vectors in Cyclic Geometric Configuration

Given the coordinates of the vertices of △ABC\triangle ABC are A(−1,2)A(-1, 2), B(1,−1)B(1, -1), and C(2,1)C(2, 1), the triangle is translated such that point AA moves to (−3,a)(-3, a) and point BB moves to (b,3)(b, 3). Find the coordinates of the point to which CC is moved.
#coordinatevectors#objectto_graph
ID: DIV-160
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 161

The symmetric point of (1, 2) with respect to the x-axis is.
#symmetry#coordinatevectors
ID: DIV-161
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 162

The coordinates of the point symmetric to point M(-1, -2, 3) with respect to the x-axis are
#symmetry#coordinatevectors
ID: DIV-162
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 163

In the Cartesian coordinate system, the coordinates of the point symmetric to point P(-1, 2) with respect to the x-axis are (　　) A: (-1, -2) B: (1, 2) C: (2, -1) D: (-2, 1)
#symmetry#coordinatevectors
ID: DIV-163
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 164

In the Cartesian coordinate system, the coordinates of the point symmetric to point P(-2, 3) with respect to the x-axis are (　　) A: (2, 3) B: (2, -3) C: (-2, -3) D: (-3, 2)
#symmetry#coordinatevectors
ID: DIV-164
Solve Problem
geometryeasy
●●○○○

Similarity Strategy · Geometry Problem 165

The coordinates of the point symmetric to the point (2, 3, 4) with respect to the x-axis are \_\_\_\_\_.
#symmetry#coordinatevectors
ID: DIV-165
Solve Problem
geometryeasy
●●○○○

Coordinates Vectors · Geometry Problem 166

In the Cartesian coordinate system, a moving point M(x, y) satisfies the condition that the moving point Q is on the curve (x−1)2+y2=1(x-1)^2+y^2=1. Then, the minimum value of ∣MQ∣|MQ| is A. B. C. 1−1- D. −-
#coordinatevectors#objectto_graph
ID: DIV-166
Solve Problem
geometryeasy
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Coordinates Vectors · Geometry Problem 167

The area of the geometric figure formed by the line obtained by shifting the graph of the function y=2x+4y=2x+4 one unit to the right and then two units up, and the two coordinate axes is (　　) A: 0 B: 16 C: 8 D: 4
#coordinatevectors#objectto_graph
ID: DIV-167
Solve Problem
geometrystretch
●●●○○

Case Analysis in Cyclic Geometric Configuration

37. a) A plane is covered with a grid of squares. Is it possible to construct an equilateral triangle with vertices at the grid points? b) In space, a regular grid of cubes is given. Is it possible to construct a regular tetrahedron with vertices coinciding with the grid points?
#case_analysis#coordinatevectors#contradiction
ID: DIV-168
Solve Problem
geometrystretch
●●●○○

Case Analysis in Cyclic Geometric Configuration

The game of Tri-Chess is played on a triangular grid. The grid is an equilateral triangle with side length nn, composed of n2n^{2} smaller equilateral triangles with side length 1. In this game, the pieces are located at the vertices of the triangles. Two rooks threaten each other if the line connecting their positions is parallel to one of the sides of the grid. For example, below, the set of vertices threatened by the rook A is shown in red; thus, rook A threatens rook B but not rook C. ![](https://cdn.mathpix.com/cropped/2024_05_10_6d86e9a87432417d5c97g-392.jpg?height=394&width=426&top_left_y=2025&top_left_x=815) When n=2022n=2022, what is the maximum number of rooks that can be placed on the grid such that no two rooks threaten each other? ## Solutions
#case_analysis#coordinatevectors#construction
ID: DIV-169
Solve Problem
geometrystretch
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Similarity Strategy · Geometry Problem 170

3. Let n≥3n \geq 3 be a natural number. How many diagonals of a regular nn-gon can be drawn such that if two diagonals intersect in the interior, they are perpendicular to each other?
#symmetry#case_analysis#coordinatevectors
ID: DIV-170
Solve Problem
mixed othercore
●●●○○

Algebraic Manipulation · Olympiad Invariants Problem 171

In the county some pairs of towns connected by two-way non-stop flight. From any town we can flight to any other (may be not on one flight). Gives, that if we consider any cyclic (i.e. beginning and finish towns match) route, consisting odd number of flights, and close all flights of this route, then we can found two towns, such that we can't fly from one to other. Proved, that we can divided all country on 44 regions, such that any flight connected towns from other regions.
#modular_arithmetic#contradiction#parity
ID: DIV-171
Solve Problem
mixed othercore
●●●○○

Case Analysis in Cyclic Geometric Configuration

Consider nn lamps clockwise numbered from 11 to nn on a circle. Let ξ\xi to be a configuration where 0≤ℓ≤n0 \le \ell \le n random lamps are turned on. A [i]cool procedure[/i] consists in perform, simultaneously, the following operations: for each one of the ℓ\ell lamps which are turned on, we verify the number of the lamp; if ii is turned on, a [i]signal[/i] of range ii is sent by this lamp, and it will be received only by the next ii lamps which follow ii, turned on or turned off, also considered clockwise. At the end of the operations we verify, for each lamp, turned on or turned off, how many signals it has received. If it was reached by an even number of signals, it remains on the same state(that is, if it was turned on, it will be turned on; if it was turned off, it will be turned off). Otherwise, it's state will be changed. The example in attachment, for n=4n=4, ilustrates a configuration where lamps 22 and 44 are initially turned on. Lamp 22 sends signal only for the lamps 33 e 44, while lamp 44 sends signal for lamps 11, 22, 33 e 44. Therefore, we verify that lamps 11 e 22 received only one signal, while lamps 33 e 44 received two signals. Therefore, in the next configuration, lamps 11 e 44 will be turned on, while lamps 22 e 33 will be turned off. Let Ψ\Psi to be the set of all 2n2^n possible configurations, where 0≤ℓ≤n0 \le \ell \le n random lamps are turned on. We define a function f:Ψ→Ψf: \Psi \rightarrow \Psi where, if ξ\xi is a configuration of lamps, then f(ξ)f(\xi) is the configurations obtained after we perform the [i]cool procedure[/i] described above. Determine all values of nn for which ff is bijective.
#case_analysis#factorization#modular_arithmetic
ID: DIV-172
Solve Problem
mixed othercore
●●●○○

Algebraic Manipulation · Olympiad Invariants Problem 173

Consider an nn-by-nn board of unit squares for some odd positive integer nn. We say that a collection CC of identical dominoes is a maximal grid-aligned configuration on the board if CC consists of (n2−1)/2(n^2-1)/2 dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: CC then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let k(C)k(C) be the number of distinct maximal grid-aligned configurations obtainable from CC by repeatedly sliding dominoes. Find the maximum value of k(C)k(C) as a function of nn.
#bounding#modular_arithmetic#contradiction
ID: DIV-173
Solve Problem
mixed othereasy
●●○○○

Objects To Graph · Olympiad Invariants Problem 174

The range of values of xx in x∈[0,2π]x \in [0,2\pi] that satisfies cos⁡x⩽12\cos x \leqslant \frac{1}{2} is (    ) A: [0,π3][0, \frac{\pi}{3}] B: [π3,2π3][\frac{\pi}{3}, \frac{2\pi}{3}] C: [π3,5π3][\frac{\pi}{3}, \frac{5\pi}{3}] D: [5π3,2π][\frac{5\pi}{3}, 2\pi]
#objectto_graph#geometry_to_algebra
ID: DIV-174
Solve Problem
mixed othereasy
●●○○○

Algebraic Manipulation · Olympiad Invariants Problem 175

Define an odd function f(x)f(x) on the interval (−∞,+∞)(-\infty,+\infty) as an increasing function, and an even function g(x)g(x) whose graph coincides with the graph of f(x)f(x) on the interval [0,+∞)[0,+\infty). Let a>b>0a>b>0, given the following inequalities: ① f(b)−f(−a)>g(a)−g(−b)f(b)-f(-a)>g(a)-g(-b) ② f(b)−f(−a)g(b)−g(−a)f(b)-f(-a)g(b)-g(-a) ④ f(a)−f(−b)<g(b)−g(−a)f(a)-f(-b)<g(b)-g(-a) Among these, the correct ones are (    ) A: ① and ④ B: ② and ③ C: ① and ③ D: ② and ④
#parity#objectto_graph
ID: DIV-175
Solve Problem
mixed othereasy
●●○○○

Algebraic Manipulation · Olympiad Invariants Problem 176

Which of the following propositions is a particular proposition? A: The graph of an even function is symmetric about the y-axis. B: All right prisms are parallelepipeds. C: Two non-intersecting lines are parallel. D: There exists a real number greater than or equal to 3.
#symmetry#parity#objectto_graph
ID: DIV-176
Solve Problem
mixed othereasy
●●○○○

Objects To Graph · Olympiad Invariants Problem 177

For \x \in [0,2\pi]\, the range of \x\ that satisfies \\cos x \leqslant \frac{1}{2}\ is \()(\quad). A: \[0, \frac{\pi}{3}]\ B: \[\frac{\pi}{3}, \frac{5\pi}{3}]\ C: \[\frac{\pi}{3}, \frac{2\pi}{3}]\ D: \[\frac{5\pi}{3},\pi]\
#objectto_graph#geometry_to_algebra
ID: DIV-177
Solve Problem
mixed othereasy
●●○○○

Algebraic Manipulation · Olympiad Invariants Problem 178

Given the inequality about xx, ∣x+a∣+∣x−1∣+a>2009|x+a|+|x-1|+a>2009 (where aa is a constant), has a non-empty set of solutions, then the range of values for aa is.
#bounding#objectto_graph
ID: DIV-178
Solve Problem
mixed othereasy
●●○○○

Algebraic Manipulation · Olympiad Invariants Problem 179

Given that proposition p: "a b, then 1a+1<1b+1\frac{1}{a+1} < \frac{1}{b+1}. Then ( ) A. "p and q" is true B. "p or q" is true C. p is false and q is true D. Both p and q are false propositions
#contradiction#objectto_graph
ID: DIV-179
Solve Problem
mixed othereasy
●●○○○

Objects To Graph · Olympiad Invariants Problem 180

The range of values of xx in [0,2π][0, 2\pi] that satisfies cos⁡x≤12\cos x \leq \frac{1}{2} is (　　) A: [0,π3][0, \frac{\pi}{3}] B: [π3,5π3][\frac{\pi}{3}, \frac{5\pi}{3}] C: [π3,2π3][\frac{\pi}{3}, \frac{2\pi}{3}] D: [5π3,π][\frac{5\pi}{3}, \pi]
#objectto_graph#geometry_to_algebra
ID: DIV-180
Solve Problem
mixed othereasy
●●○○○

Objects To Graph · Olympiad Invariants Problem 181

In the interval \(0,2\pi)\, the range of \x\ that satisfies \\sin x > \cos x\ is \()(\quad). A: \( \dfrac {\pi}{4}, \dfrac {\pi}{2})\cup(\pi, \dfrac {5\pi}{4})\ B: \( \dfrac {\pi}{4},\pi)\ C: \( \dfrac {\pi}{4},\pi)\cup( \dfrac {5\pi}{4}, \dfrac {3\pi}{2})\ D: \( \dfrac {\pi}{4}, \dfrac {5\pi}{4})\
#objectto_graph#geometry_to_algebra
ID: DIV-181
Solve Problem
mixed othereasy
●●○○○

Algebraic Manipulation · Olympiad Invariants Problem 182

Let the variables xx and yy satisfy ∣x∣+∣y∣≤1|x| + |y| \leq 1. The maximum value of x+2yx + 2y is \_\_\_\_\_\_\_\_.
#construction#objectto_graph
ID: DIV-182
Solve Problem
mixed othereasy
●●○○○

Algebraic Manipulation · Olympiad Invariants Problem 183

If the solution set of the inequality ax2+bx+c>0ax^2+bx+c>0 (where a>0a>0) is R\mathbb{R}, then the correct conclusion among the following is ( ) A: b2−4ac>0b^2-4ac>0 B: b2−4ac<0b^2-4ac<0 C: b2−4ac≤0b^2-4ac\leq0 D: b2−4ac≥0b^2-4ac\geq0
#bounding#objectto_graph
ID: DIV-183
Solve Problem
mixed othereasy
●●○○○

Coordinates Vectors · Olympiad Invariants Problem 184

A robot cat moves forward or backward one step per second, programmed to move three steps forward and then two steps backward in a pattern. If this robot cat is placed at the origin of the number line, facing the positive direction, and moves one step as one unit of distance. Let P(n)P(n) represent the coordinate of the robot cat's position at the nn-th second, and P(0)=0P(0) = 0, then among the following statements, the incorrect one is ( ) A: P(3)=3P(3) = 3 B: P(5)=1P(5) = 1 C: P(101)=21P(101) = 21 D: P(101)>P(104)P(101) > P(104)
#coordinatevectors#change_representation
ID: DIV-184
Solve Problem
mixed othereasy
●●○○○

Case Analysis · Olympiad Invariants Problem 185

Let M={x∣x2−x>0}M=\{x|x^2-x > 0\} and N={x∣1x<1}N=\{x| \frac{1}{x} < 1\}. Then A: M⊊NM \subsetneq N B: N⊊MN \subsetneq M C: M=NM = N D: M∪N=RM \cup N = \mathbb{R}
#case_analysis#bounding#factorization
ID: DIV-185
Solve Problem
mixed othereasy
●●○○○

Algebraic Manipulation & Combinatorial Partitions

For any x∈Rx \in \mathbb{R}, the inequality ax2+ax−1<0ax^2+ax-1<0 always holds. Then, the range of aa is ( ) A: a≤0a\leq0 B: a<4a<4 C: −4<a<0-4<a<0 D: −4<a≤0-4<a\leq0
#bounding#objectto_graph
ID: DIV-186
Solve Problem
mixed othereasy
●●○○○

Algebraic Manipulation · Olympiad Invariants Problem 187

Which of the following propositions is a particular proposition? A: The graph of an even function is symmetric about the y-axis. B: All square prisms are parallelepipeds. C: Two non-intersecting lines are parallel. D: There exists a real number greater than or equal to 3.
#symmetry#parity#objectto_graph
ID: DIV-187
Solve Problem
mixed othereasy
●●○○○

Coordinates Vectors · Olympiad Invariants Problem 188

A robot moves one step forward or backward every second. The programmer designed the robot to move forward 3 steps and then move backward 2 steps in a pattern. If the robot is placed at the origin of the number line, facing the positive direction, with a step distance of 1 unit length. Let P(n)P(n) represent the coordinate of the robot's position at the nn-th second, and let P(0)=0P(0) = 0. Among the following statements, the incorrect one is ( ) A: P(3)=3P(3) = 3 B: P(5)=1P(5) = 1 C: P(2003)>P(2005)P(2003) > P(2005) D: P(2003)<P(2005)P(2003) < P(2005)
#coordinatevectors#change_representation
ID: DIV-188
Solve Problem
mixed othereasy
●●○○○

Case Analysis & Combinatorial Partitions

The proposition "For all x∈Rx \in \mathbb{R}, ax2−2ax+3>0ax^{2}-2ax+3 > 0 always holds true" is a true proposition. The range of values for the real number aa is ______\_\_\_\_\_\_.
#case_analysis#objectto_graph
ID: DIV-189
Solve Problem
mixed otherstretch
●●●○○

Case Analysis & Combinatorial Partitions

C4. Let kk and ss be positive integers such that s<(2k+1)2s<(2 k+1)^{2}. Initially, one cell out of an n×nn \times n grid is coloured green. On each turn, we pick some green cell cc and colour green some ss out of the (2k+1)2(2 k+1)^{2} cells in the (2k+1)×(2k+1)(2 k+1) \times(2 k+1) square centred at cc. No cell may be coloured green twice. We say that ss is kk-sparse if there exists some positive number CC such that, for every positive integer nn, the total number of green cells after any number of turns is always going to be at most CnC n. Find, in terms of kk, the least kk-sparse integer ss. ## NIKOLAI BELUHOV
#case_analysis#bounding#coordinatevectors
ID: DIV-190
Solve Problem
number theorycore
●●●○○

Coordinates Vectors · Number Theory Problem 191

A 150×324×375150\times 324\times 375 [rectangular](https://artofproblemsolving.com/wiki/index.php/Rectangle) [solid](https://artofproblemsolving.com/wiki/index.php/Solid) is made by gluing together 1×1×11\times 1\times 1 cubes. An internal [diagonal](https://artofproblemsolving.com/wiki/index.php/Diagonal) of this solid passes through the interiors of how many of the 1×1×11\times 1\times 1 [ cubes](https://artofproblemsolving.com/wiki/index.php/Cube_(geometry))?
#coordinatevectors#gcd_structure#change_representation
ID: DIV-191
Solve Problem
number theorycore
●●●○○

Case Analysis · Number Theory Problem 192

A sequence a1,a2,…a_1,a_2,\dots of non-negative integers is defined by the rule an+2=∣an+1−an∣a_{n+2}=|a_{n+1}-a_n| for n≥1n\geq 1. If a1=999a_1=999, a2<999a_2<999 and a2006=1a_{2006}=1, how many different values of a2a_2 are possible? (A) 165(B) 324(C) 495(D) 499(E) 660\mathrm{(A)}\ 165 \qquad \mathrm{(B)}\ 324 \qquad \mathrm{(C)}\ 495 \qquad \mathrm{(D)}\ 499 \qquad \mathrm{(E)}\ 660
#case_analysis#modular_arithmetic#contradiction
ID: DIV-192
Solve Problem
number theorycore
●●●○○

Case Analysis & Modular Valuation

(Titu Andreescu) Prove that for every positive integer nn there exists an nn-digit number divisible by 5n5^n all of whose digits are odd.
#case_analysis#modular_arithmetic#contradiction
ID: DIV-193
Solve Problem
number theorycore
●●●○○

Modular Strategy · Number Theory Problem 194

You are given n≥2n \ge 2 distinct positive integers. Let's call a pair of these integers [i]elegant[/i] if their sum is an integer power of 22. For every nn find the largest possible number of elegant pairs. [i]Proposed by Oleksiy Masalitin[/i]
#contradiction#construction#change_representation
ID: DIV-194
Solve Problem
number theorycore
●●●○○

Case Analysis & Modular Valuation

Some pairs of cities in a certain country are connected by roads, at least three roads going out of each city. Prove that there exists a round path consisting of roads whose number is not divisible by 33.
#case_analysis#modular_arithmetic#contradiction
ID: DIV-195
Solve Problem
number theorycore
●●●○○

Modular Strategy · Number Theory Problem 196

Theorem 2 Let a,b,ca, b, c be integers, a>0,ac−b2=1a>0, a c-b^{2}=1. Then the equation ax2+2bxy+cy2=1a x^{2}+2 b x y+c y^{2}=1 has integer solutions.
#symmetry#bounding#factorization
ID: DIV-196
Solve Problem
number theorycore
●●●○○

Modular Strategy · Number Theory Problem 197

Let N\mathbb{N} denote the set of positive integers. Find all functions f :N→Nf : \mathbb{N} \rightarrow \mathbb{N} such that for positive integers aa and b,b,f(a2+b2)=f(a)f(b) and f(a2)=f(a)2.f(a^2 + b^2) = f(a)f(b) \text{ and } f(a^2) = f(a)^2.
#factorization#modular_arithmetic#parity
ID: DIV-197
Solve Problem
number theorycore
●●●○○

Case Analysis in Cyclic Geometric Configuration

(Zuming Feng) Determine all composite positive integers nn for which it is possible to arrange all divisors of nn that are greater than 1 in a circle so that no two adjacent divisors are relatively prime.
#case_analysis#factorization#construction
ID: DIV-198
Solve Problem
number theoryeasy
●●○○○

Coordinates Vectors · Number Theory Problem 199

If point P (x, y) has both its x-coordinate and y-coordinate as integers, then point P is called an integer point. In the square with vertices at (10, 0), (0, 10), (-10, 0), and (0, -10) (including the boundary), the total number of integer points is (　　) A: 220 B: 221 C: 222 D: 223
#coordinatevectors#objectto_graph
ID: DIV-199
Solve Problem
number theoryeasy
●●○○○

Coordinates Vectors · Number Theory Problem 200

A robotic cat moves forward or backward one step per second, programmed to move three steps forward and then two steps backward in sequence. If the robotic cat starts at the origin of the number line, facing the positive direction, with a step distance of one unit, let P(n) represent the coordinate of the robotic cat's position at the n-th second, and P(0) = 0, then among the following conclusions, the correct one is ( ) A: P(1) = 3 B: P(5) = 1 C: P(2009) < P(2010) D: P(2009) = 401
#coordinatevectors#change_representation
ID: DIV-200
Solve Problem
number theoryeasy
●●○○○

Modular Strategy · Number Theory Problem 201

In the Cartesian coordinate system, there are several points with integer abscissas, in the order of (0,0)\left(0,0\right), (1,1)\left(1,1\right), (2,0)\left(2,0\right), (3,−1)\left(3,-1\right), (4,0)\left(4,0\right), (5,1)\left(5,1\right), (6,0)\left(6,0\right), (7,−1)\left(7,-1\right), (8,0)\left(8,0\right), …\ldots , according to this pattern, the coordinates of the 20232023rd point are ______.
#modular_arithmetic#coordinatevectors
ID: DIV-201
Solve Problem
number theoryeasy
●●○○○

Modular Strategy & Modular Valuation

The famous Goldbach Conjecture states: "Every even number greater than 22 can be expressed as the sum of two prime numbers." To study this conjecture using proof by contradiction, the assumed content should be ____.
#contradiction#parity
ID: DIV-202
Solve Problem
number theoryeasy
●●○○○

Case Analysis · Number Theory Problem 203

Given the recursive formula of the sequence an{a_n} as follows: an={n,if n is oddan2,if n is even(n∈N∗)a_n = \begin{cases} n,& \text{if } n \text{ is odd} \\ a_{\frac{n}{2}}, & \text{if } n \text{ is even} \end{cases} (n \in \mathbb{N}^*), find the sum of a48+a49=‾a_{48} + a_{49} = \underline{\hspace{1cm}} and determine the position of the 9th occurrence of 5 in the sequence.
#case_analysis#parity#recursion
ID: DIV-203
Solve Problem
number theoryeasy
●●○○○

Case Analysis · Number Theory Problem 204

We can use the recursive formula of the sequence {an}\{a_n\}, an={n,when n is oddan2,when n is even(n∈N+)a_n=\begin{cases}n, & \text{when } n \text{ is odd} \\ a_{\frac{n}{2}}, & \text{when } n \text{ is even}\end{cases} (n\in\mathbb{N}^{+}), to find the values of the terms in this sequence, so that each term in the sequence is an odd number. Then, the value of a64+a65=a_{64}+a_{65}= \_\_\_\_\_\_.
#case_analysis#parity#recursion
ID: DIV-204
Solve Problem
number theoryeasy
●●○○○

Case Analysis · Number Theory Problem 205

Given an even function f(x) ⁣:Z→Zf(x)\colon \mathbb{Z} \to \mathbb{Z} satisfying f(1)=1f(1)=1, f(2015)≠1f(2015) \neq 1, and for any integers aa, bb, we have f(a+b)≤max⁡{f(a),f(b)}f(a+b) \leq \max\{f(a),f(b)\}, where max⁡(x,y)={xif x≥yyif x<y\max(x,y)= \begin{cases} x & \text{if } x \geq y \\ y & \text{if } x < y \end{cases} Determine the value of f(2016)f(2016). Options: A: 0 B: 1 C: 2015 D: 2016
#case_analysis#contradiction#parity
ID: DIV-205
Solve Problem
number theoryeasy
●●○○○

Modular Strategy · Number Theory Problem 206

For any two positive integers, define an operation mΔnm \Delta n: when both mm and nn are either positive even numbers or positive odd numbers, mΔn=m+nm \Delta n = m + n; when one of mm and nn is a positive odd number and the other is a positive even number, mΔn=mnm \Delta n = mn. Under the definition above, let M={(x,y)∣xΔy=36,x∈N∗,y∈N∗}M = \{(x,y) | x \Delta y = 36, x \in \mathbb{N}^*, y \in \mathbb{N}^*\}. The number of elements in the set MM is A: 40 B: 48 C: 39 D: 41
#factorization#parity
ID: DIV-206
Solve Problem
number theoryeasy
●●○○○

Modular Strategy · Number Theory Problem 207

Gauss is a famous German mathematician, known as the "Prince of Mathematics". There are 110 achievements named after his name "Gauss". For x∈Rx\in R, let [x][x] represent the largest integer not greater than xx, and let {x}=x−[x]\{x\}=x-[x] represent the non-negative fractional part of xx. Then, y=[x]y=[x] is called the Gauss function. It is known that the sequence {an}\{a_{n}\} satisfies: a1=3a_{1}=\sqrt{3}, an+1=[an]+1{an}a_{n+1}=[a_{n}]+\frac{1}{\{a_{n}\}}, (n∈N∗)(n∈N^{*}), then a2019=a_{2019}=\_\_\_\_\_\_\_\_.
#parity#recursion#change_representation
ID: DIV-207
Solve Problem
number theoryeasy
●●○○○

Modular Strategy · Number Theory Problem 208

Let ana_n denote the largest odd factor of the natural number nn. For example, the factors of 99 are 11, 33, and 99, so a9=9a_9 = 9; the factors of 1010 are 11, 22, 55, and 1010, so a10=5a_{10} = 5. Let SnS_n denote the sum of the first nn terms of the sequence an{a_n}. Find S22016−1S_{2^{2016} - 1}.
#factorization#parity#recursion
ID: DIV-208
Solve Problem
number theoryeasy
●●○○○

Modular Strategy · Number Theory Problem 209

If 50a\sqrt{50a} is an integer, then the smallest positive integer a=a = ____.
#factorization#polynomial_roots
ID: DIV-209
Solve Problem
number theoryeasy
●●○○○

Modular Strategy · Number Theory Problem 210

Let [m]\left[m\right] represent the integer part of a real number mm, for example, [23]=0,[3.14]=3[\frac{2}{3}]=0, \left[3.14\right]=3. According to this rule, the value of [−17+1][-\sqrt{17}+1] is ______.
#bounding#change_representation#polynomial_roots
ID: DIV-210
Solve Problem
number theoryeasy
●●○○○

Modular Strategy & Modular Valuation

Write down the greatest common divisor (GCD) and the least common multiple (LCM) for each pair of numbers below: 6 and 12; 7 and 8; 15 and 20.
#factorization#gcd_structure
ID: DIV-211
Solve Problem
number theoryeasy
●●○○○

Case Analysis · Number Theory Problem 212

The "Collatz Conjecture," also known as the "3n+1 Conjecture," is a conjecture proposed by German mathematician Lothar Collatz at the 1950 World Mathematics Congress: For any given positive integer n, if n is even, it is halved; if n is odd, it is multiplied by 3 and added by 1. Repeat this operation indefinitely, and eventually, it will reach 1. A positive integer m becomes 1 after 6 iterations. What is the value of m?
#case_analysis#parity#transformation
ID: DIV-212
Solve Problem
number theoryeasy
●●○○○

Modular Strategy · Number Theory Problem 213

We use the symbol [x]\left[x\right] to represent the largest integer that is not greater than the real number xx. For example, [3.69]=3\left[3.69\right]=3, [−0.56]=−1\left[-0.56\right]=-1. According to this rule, [−5−1]=[-\sqrt{5}-1]=____.
#bounding#change_representation#polynomial_roots
ID: DIV-213
Solve Problem
number theoryeasy
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Modular Strategy · Number Theory Problem 214

Given that 96n\sqrt{96n} is an integer, the smallest value of the positive integer nn is ( ) A: 9696 B: 66 C: 2424 D: 22
#factorization#polynomial_roots
ID: DIV-214
Solve Problem
number theoryeasy
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Case Analysis · Number Theory Problem 215

The mathematicians of the ancient Greek Pythagorean school studied various polygonal numbers, such as triangular numbers (1)(1), (3)(3), (6)(6), (10)(10), (…)(…), with the nn-th triangular number being n(n+1)2=12n2+12n\dfrac {n(n+1)}{2}= \dfrac {1}{2}n^{2}+ \dfrac {1}{2}n. Let N(n,k)N(n,k) represent the nn-th kk-sided polygonal number (k⩾3)(k\geqslant 3). The following lists some expressions for the nn-th kk-sided polygonal number: Triangular numbers N(n,3)=12n2+12nN(n,3)= \dfrac {1}{2}n^{2}+ \dfrac {1}{2}n Square numbers N(n,4)=n2N(n,4)=n^{2} Pentagonal numbers N(n,5)=32n2−12nN(n,5)= \dfrac {3}{2}n^{2}- \dfrac {1}{2}n Hexagonal numbers N(n,6)=2n2−nN(n,6)=2n^{2}-n …… We can conjecture the expression for N(n,k)N(n,k) and use it to calculate N(8,12)=N(8,12)= _____ .
#case_analysis#change_representation#induction
ID: DIV-215
Solve Problem
number theoryeasy
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Coordinates Vectors · Number Theory Problem 216

The distance from point AA on the number line to the origin is 66. What number does point AA represent? A: 66 B: −6-6 C: 66 or −6-6 D: 33 or −3-3
#coordinatevectors#change_representation
ID: DIV-216
Solve Problem
number theoryeasy
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Case Analysis & Combinatorial Partitions

Given a sequence {a_n}\{a\_n\} where all terms are positive integers, for n=1n=1, 22, 33, ......, we have an+1={3an+5an is oddan2kan is even. wherek is the positive integer making an+1 odda_{n+1}= \begin{cases}3a_{n}+5 & a_{n}\text{ is odd} \\\\ \frac {a_{n}}{2^{k}} & a_{n}\text{ is even. }{where }k\text{ is the positive integer making }a_{n+1}\text{ odd}\end{cases}, when a1=11a_{1}=11, a100=a_{100}= _______ ; If there exists m∈N∗m∈N^{*}, when n>mn > m and ana_{n} is odd, ana_{n} is always a constant pp, then the value of pp is _______ .
#case_analysis#parity#recursion
ID: DIV-217
Solve Problem
number theoryeasy
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Case Analysis in Cyclic Geometric Configuration

Let the function f(x)={1,x∈Q0,x∈R∖Qf(x)= \begin{cases} 1, & x\in\mathbb{Q} \\ 0, & x\in\mathbb{R}\setminus\mathbb{Q}\end{cases}, where R\mathbb{R} is the set of real numbers, and Q\mathbb{Q} is the set of rational numbers. Regarding the function f(x)f(x), there are the following four propositions: (1)f(f(x))=0(1) f(f(x))=0; (2)(2) The function f(x)f(x) is an even function; (3)(3) For any non-zero rational number TT, f(x+T)=f(x)f(x+T)=f(x) holds true for any xx; (4)(4) There are at least three points AA, BB, CC on the graph of the function f(x)f(x), such that △ABC\triangle ABC is an equilateral triangle. Among these, the true propositions are \_\_\_\_\_\_ (Write down the numbers of all true propositions).
#case_analysis#parity#objectto_graph
ID: DIV-218
Solve Problem
number theoryeasy
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Modular Strategy · Number Theory Problem 219

When nn is a positive integer, define the function N(n)N(n) to represent the largest odd factor of nn. For example, N(3)=3N(3)=3, N(10)=5N(10)=5, etc. Let S(n)=N(1)+N(2)+N(3)+…+N(2n).S(n)=N(1)+N(2)+N(3)+…+N(2^{n}). Then, (1)S(4)=(1)S(4)= \_\_\_\_\_\_ (2)S(n)=(2)S(n)= \_\_\_\_\_\_ .
#factorization#parity#change_representation
ID: DIV-219
Solve Problem
number theoryeasy
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Case Analysis · Number Theory Problem 220

For natural numbers greater than 11, the cube of 2×22 \times 2 can be "split" into odd numbers in the following manner: 23{53,  33{7911,  43{13151719,…2^{3} \begin{cases} \overset{3}{5}\end{cases}, \;3^{3} \begin{cases} 7 \\ 9 \\ 11\end{cases}, \;4^{3} \begin{cases} 13 \\ 15 \\ 17 \\ 19\end{cases}, \ldots If one of the "splitting numbers" of m3m^{3} is 345345, then mm is (    )(\ \ \ \ ) A: 1616 B: 1717 C: 1818 D: 1919
#case_analysis#parity#induction
ID: DIV-220
Solve Problem
number theoryeasy
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Case Analysis in Cyclic Geometric Configuration

Given the function f(x)={1,x is rational0,x is irrationalf(x)= \begin{cases} 1,x{\text{ is rational}}\\ 0,x{\text{ is irrational}} \end{cases}, there are four propositions: 1. For any x∈Rx∈R, f(f(x))=1f(f(x))=1; 2. The function f(x)f(x) is an even function; 3. If TT is a non-zero rational number, then f(x+T)=f(x)f(x+T)=f(x) holds for any x∈Rx∈R; 4. There are three points AA, BB, CC on the graph of f(x)f(x) such that △ABC△ABC is an equilateral triangle. The sequence number of the correct propositions is _______.
#case_analysis#parity#objectto_graph
ID: DIV-221
Solve Problem
number theoryeasy
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Modular Strategy · Number Theory Problem 222

Gauss is a famous German mathematician, known as the "Prince of Mathematics". There are 110 achievements named after "Gauss". Suppose x∈Rx\in R, use [x][x] to represent the largest integer not exceeding xx, and use {x}=x−[x]\{x\}=x-[x] to represent the non-negative fractional part of xx. Then y=[x]y=[x] is called the Gauss function. It is known that the sequence {an}\{a_{n}\} satisfies: a1=3,an+1=[an]+1{an},(n∈N∗)a_{1}= \sqrt {3},a_{n+1}=[a_{n}]+ \frac {1}{\{a_{n}\}},(n∈N^{*}) , then a2017=a_{2017}= ___ .
#recursion#change_representation#induction
ID: DIV-222
Solve Problem
number theoryeasy
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Case Analysis · Number Theory Problem 223

The sequence {an}\{a_n\} satisfies an={an−12,if an−1 is even3an−1+1,if an−1 is odd a_n= \begin{cases} \frac{a_{n-1}}{2}, & \text{if } a_{n-1} \text{ is even} \\ 3a_{n-1}+1, & \text{if } a_{n-1} \text{ is odd} \end{cases} If a1=34a_1=34, then the sum of the first 100100 terms of the sequence {an}\{a_n\} is ______.
#case_analysis#parity#recursion
ID: DIV-223
Solve Problem
number theoryeasy
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Case Analysis · Number Theory Problem 224

For natural numbers mm greater than 1, the cube of mm can be "split" into odd numbers in the following way: 23{352^{3} \begin{cases} 3 \\ 5\end{cases}, 33{79113^{3} \begin{cases} 7 \\ 9 \\ 11\end{cases}, 43{131517194^{3} \begin{cases} 13 \\ 15 \\ 17 \\ 19\end{cases}... In this manner, if one of the "split numbers" of m3m^3 is 59, then the value of mm is \_\_\_\_\_\_.
#case_analysis#bounding#parity
ID: DIV-224
Solve Problem
number theoryeasy
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Modular Strategy · Number Theory Problem 225

(1) Are there any integer solutions xx and yy that satisfy the equation x2−y2=1998x^2 - y^2 = 1998? If yes, find the solutions; if no, explain why. (2) If the vertices of a cube are labeled with +1 or -1, and each face is labeled with a number equal to the product of the numbers at its four vertices, can the sum of the 14 labeled numbers be 0?
#factorization#parity#objectto_graph
ID: DIV-225
Solve Problem
number theoryeasy
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Case Analysis · Number Theory Problem 226

Given that the cube of a positive integer mm can be "split" into odd numbers in the following manner: 131,23begincases35endcases,33begincases7911endcases,43begincases13151719endcases)andsoon.Ifthe"split"of1^{3}\\{1,2^{3} \\begin{cases} 3 \\\\ 5\\end{cases},3^{3} \\begin{cases} 7 \\\\ 9 \\\\ 11\\end{cases},4^{3} \\begin{cases} 13 \\\\ 15 \\\\ 17 \\\\ 19\\end{cases}\\}\\) and so on. If the "split" of m^{3}containstheoddnumber contains the odd number 2015,whatisthevalueof, what is the value of m$?
#case_analysis#parity#induction
ID: DIV-226
Solve Problem
number theoryeasy
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Modular Strategy in Cyclic Geometric Configuration

The ancient Chinese mathematical masterpiece "Zhou Bi Suan Jing" records the phrase "multiply the legs and the base of a right triangle by themselves, add them together, and then take the square root of the sum," which is symbolically represented as a2+b2=c2(a,b,c∈N∗)a^{2}+b^{2}=c^{2} (a,b,c \in \mathbb{N}^{*}). We call aa, bb, and cc Pythagorean triples. Given several sets of Pythagorean triples: 3,4,53, 4, 5; 5,12,135, 12, 13; 7,24,257, 24, 25; 9,40,419, 40, 41, and so on, we can guess that the three numbers of the fifth set of Pythagorean triples are ______.
#parity#polynomial_roots#induction
ID: DIV-227
Solve Problem
number theoryeasy
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Case Analysis · Number Theory Problem 228

For natural numbers mm greater than 11, the cube of mm can be "split" into odd numbers in the following manner: 232^{3} yields {35\begin{cases} 3 \\ 5\end{cases}, 333^{3} yields {7911\begin{cases} 7 \\ 9 \\ 11\end{cases}, 434^{3} yields {13151719\begin{cases} 13 \\ 15 \\ 17 \\ 19\end{cases}, and so on. Following this pattern, if one of the "split numbers" of m3m^{3} is 5959, then the value of mm is \_\_\_\_\_\_.
#case_analysis#bounding#parity
ID: DIV-228
Solve Problem
number theoryeasy
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Modular Strategy · Number Theory Problem 229

When nn is a positive integer, define the function N(n)N(n) to represent the largest odd factor of nn. For example, N(3)=3N(3)=3, N(10)=5N(10)=5, ..., S(n)=N(1)+N(2)+N(3)+…+N(2n)S(n)=N(1)+N(2)+N(3)+…+N(2^{n}), then S(5)=S(5)= 　　() A: 342342 B: 345345 C: 341341 D: 346346
#factorization#parity#change_representation
ID: DIV-229
Solve Problem
number theoryeasy
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Modular Strategy · Number Theory Problem 230

Please write down a positive integer value of mm such that 8m\sqrt{8m} is an integer: m=______.m=\_\_\_\_\_\_.
#factorization#parity#polynomial_roots
ID: DIV-230
Solve Problem
number theoryeasy
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Modular Strategy · Number Theory Problem 231

In a card flipping game during a math class, the teacher numbered 54 playing cards in the order of 1, 2, 3, …, 54 and placed them face down in a row. There were exactly 54 students in the class, who were also numbered in the order of 1, 2, 3, …, 54. The rules of the game were as follows: the student numbered 1 flipped over all the cards that were multiples of 1; the student numbered 2 flipped over all the cards that were multiples of 2; the student numbered 3 flipped over all the cards that were multiples of 3, and so on, until the last 54th student flipped over card number 54 to end the game. The question is: how many playing cards were face up at the end of the game? List their numbers and explain the reason.
#factorization#parity
ID: DIV-231
Solve Problem
number theoryeasy
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Case Analysis · Number Theory Problem 232

Given any two positive integers, a certain operation (denoted by the operator ⊕\oplus) is defined as follows: when mm and nn are both positive even numbers or both positive odd numbers, m⊕n=m+nm \oplus n = m + n; when one of mm and nn is a positive even number and the other is a positive odd number, m⊕n=m⋅nm \oplus n = m \cdot n. The number of elements in the set M=(a,b)∣a⊕b=12,a,b∈N∗M = {(a, b) \mid a \oplus b = 12, a, b \in \mathbb{N}^*} is ______\_\_\_\_\_\_.
#case_analysis#parity#change_representation
ID: DIV-232
Solve Problem
number theoryeasy
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Case Analysis in Cyclic Geometric Configuration

Given the function f(x)={1,if x is rational0,if x is irrationalf(x) = \begin{cases} 1, & \text{if } x \text{ is rational} \\ 0, & \text{if } x \text{ is irrational} \end{cases}, consider the following three statements: ① The function f(x)f(x) is an even function; ② The function f(x)f(x) is a periodic function; ③ There exist xix_i (i=1,2,3i=1,2,3), such that the triangle with vertices (xi,f(xi))(x_i, f(x_i)) is an equilateral triangle. The number of correct statements is (　　) A: 0 B: 1 C: 2 D: 3
#case_analysis#parity#objectto_graph
ID: DIV-233
Solve Problem
number theoryeasy
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Modular Strategy & Modular Valuation

The greatest common divisor of the three numbers 390, 455, and 546 is ______.
#factorization#gcd_structure
ID: DIV-234
Solve Problem
number theoryeasy
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Modular Strategy · Number Theory Problem 235

If the integer part of a3\sqrt[3]{a} is 22, then the number of odd numbers aa that satisfy this condition is ____.
#bounding#parity#polynomial_roots
ID: DIV-235
Solve Problem
number theoryeasy
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Coordinates Vectors · Number Theory Problem 236

A robotic cat moves forward or backward by one step every second. The programmer has designed the robotic cat to move forward 3 steps and then move backward 2 steps in a cycle. If this robotic cat is placed at the origin of the number line, facing the positive direction, with a step distance of 1 unit length, let P(n) represent the coordinate of the position of the robotic cat at the nth second, and P(0) = 0, then among the following conclusions, the incorrect one is ( ) A: P(3) = 3 B: P(5) = 1 C: P(101) = 21 D: P(103) < P(104)
#coordinatevectors#change_representation
ID: DIV-236
Solve Problem
number theoryeasy
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Case Analysis & Modular Valuation

The smallest natural number nn that has exactly 10 positive divisors.
#case_analysis#factorization
ID: DIV-237
Solve Problem
number theoryeasy
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Case Analysis · Number Theory Problem 238

Observation: Because 4<5<9\sqrt{4} \lt \sqrt{5} \lt \sqrt{9}, that is, 2<5<32 \lt \sqrt{5} \lt 3, so the integer part of 5\sqrt{5} is 22, and the decimal part is 5−2\sqrt{5}-2. Please solve the following problem after observing the above pattern: (1)(1) Define the symbol [m]\left[m\right] to represent the integer part of a real number mm, for example: [23]=0[\frac{2}{3}]=0, [6]=2[\sqrt{6}]=2. According to this rule, what is the value of [10+1][\sqrt{10}+1]? (2)(2) If the integer part of 11\sqrt{11} is aa, the decimal part is bb, and ∣c∣=11|c|=\sqrt{11}, find the value of c(a−b−6)+12c\left(a-b-6\right)+12.
#case_analysis#bounding#change_representation
ID: DIV-238
Solve Problem
number theoryeasy
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Case Analysis · Number Theory Problem 239

We can use the recursive formula of the sequence {an}\{a_n\}: an={n,when n is oddan2,when n is even a_n = \begin{cases} n, & \text{when } n \text{ is odd} \\ a_{\frac{n}{2}}, & \text{when } n \text{ is even} \end{cases} (n∈N∗n \in \mathbb{N}^*) to find the values of the terms in this sequence, ensuring that each term in the sequence is an odd number. Then, a48+a49=______a_{48} + a_{49} = \_\_\_\_\_\_; it has been found that the odd numbers in this sequence will repeat, so the ninth occurrence of 5 is the ______\_\_\_\_\_\_ term of this sequence.
#case_analysis#parity#recursion
ID: DIV-239
Solve Problem
number theoryeasy
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Modular Strategy · Number Theory Problem 240

Given that the product of four distinct integers aa, bb, cc, dd is 1414, then their sum is ( ) A: −5-5 B: 55 C: 99 D: 55 or −5-5
#factorization#parity
ID: DIV-240
Solve Problem
number theoryeasy
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Case Analysis · Number Theory Problem 241

Is there a pair of positive integers xx and yy such that x2=y2+2023x^{2}=y^{2}+2023? If such pair exists, find the values of xx and yy; if not, please explain the reason.
#case_analysis#factorization
ID: DIV-241
Solve Problem
number theoryeasy
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Case Analysis · Number Theory Problem 242

For any natural number mm greater than 11, the cube of mm can be "split" into consecutive odd numbers as follows: 2^{3}= \begin{cases} 3 \\\\ 5\\end{cases}, 3^{3}= \begin{cases} 7 \\\\ 9 \\\\ 11\\end{cases}, 4^{3}= \begin{cases} 13 \\\\ 15 \\\\ 17 \\\\ 19\\end{cases}, (...)(...) Following this pattern, if the "split" of m3m^{3} contains the number 20152015, then m=m= _____ .
#case_analysis#parity#induction
ID: DIV-242
Solve Problem
number theoryeasy
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Modular Strategy · Number Theory Problem 243

For any two positive integers, define the operation ⊕\oplus as follows: when mm and nn are both positive even numbers or both positive odd numbers, m⊕n=m+nm \oplus n = m + n; when one of mm and nn is a positive even number and the other is a positive odd number, m⊕n=m×nm \oplus n = m \times n. For example, 4⊕6=4+6=104 \oplus 6 = 4 + 6 = 10, 3⊕7=3+7=103 \oplus 7 = 3 + 7 = 10, 3⊕4=3×4=123 \oplus 4 = 3 \times 4 = 12. Under this definition, the set M={(a,b)∣a⊕b=12,a,b∈N+}M = \{(a, b) \mid a \oplus b = 12, a, b \in \mathbb{N}^+\} has ___ elements.
#symmetry#case_analysis#parity
ID: DIV-243
Solve Problem
number theoryeasy
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Modular Strategy · Number Theory Problem 244

Given that 12n\sqrt{12n} is an integer, find the smallest positive integer value of nn.
#factorization#parity#polynomial_roots
ID: DIV-244
Solve Problem
number theoryeasy
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Modular Strategy & Modular Valuation

The greatest common divisor (GCD) of the three numbers 72, 120, and 168 is _____.
#factorization#gcd_structure
ID: DIV-245
Solve Problem
number theoryeasy
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Coordinates Vectors & Modular Valuation

For the following propositions, which one is a true statement? A: All prime numbers are odd numbers B: For all x∈{y∣yx\in \{y\left|\right.y is an irrational number}\}, x3x^{3} is an irrational number C: In the Cartesian coordinate system, at least one graph of a quadratic function does not intersect the yy-axis D: The negation of the proposition "There is at least one integer nn such that n2+nn^{2}+n is odd"
#coordinatevectors#parity#objectto_graph
ID: DIV-246
Solve Problem
number theoryeasy
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Modular Strategy · Number Theory Problem 247

The Dayan sequence, originating from the "Qiankun Pu" in the interpretation of the "Great Expansion Number Fifty" in the Yi Jing, is mainly used to explain the principle of Taiji derivation in traditional Chinese culture. Each term in the sequence represents the total number of the two states that have been experienced in the process of Taiji derivation. It is the first sequence problem in the history of world mathematics hidden in Chinese traditional culture. The first 10 terms are 0, 2, 4, 8, 12, 18, 24, 32, 40, 50…, then the 20th term of this sequence is ( ) A: 180 B: 200 C: 128 D: 162
#parity#recursion#induction
ID: DIV-247
Solve Problem
number theoryshowcase
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Case Analysis & Modular Valuation

The rows and columns of a 2n×2n2^{n} \times 2^{n} table are numbered from 0 to 2n−12^{n}-1. The cells of the table have been colored with the following property being satisfied: for each 0≤i,j≤2n−10 \leq i, j \leq 2^{n}-1, the jj th cell in the ii th row and the (i+j)(i+j) th cell in the jj th row have the same color. (The indices of the cells in a row are considered modulo 2n2^{n}.) Prove that the maximal possible number of colors is 2n2^{n}.
#case_analysis#factorization#modular_arithmetic
ID: DIV-248
Solve Problem
number theorystretch
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Modular Strategy & Modular Valuation

N8 Let kk be a positive integer and set n=2k+1n=2^{k}+1. Prove that nn is a prime number if and only if the following holds: there is a permutation a1,…,an−1a_{1}, \ldots, a_{n-1} of the numbers 1,2,…,n−11,2, \ldots, n-1 and a sequence of integers g1,g2,…,gn−1g_{1}, g_{2}, \ldots, g_{n-1} such that nn divides giai−ai+1g_{i}^{a_{i}}-a_{i+1} for every i∈{1,2,…,n−1}i \in\{1,2, \ldots, n-1\}, where we set an=a1a_{n}=a_{1}.
#factorization#modular_arithmetic#contradiction
ID: DIV-249
Solve Problem
number theorystretch
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Modular Strategy · Number Theory Problem 250

Let's calculate how many two-digit numbers there are in the base nn number system, the sum of whose digits is also two-digit (in the base nn system). Give the result in the base nn number system.
#symmetry#coordinatevectors#parity
ID: DIV-250
Solve Problem