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4,079 Curated Olympiad Problems · 65 Capabilities · 419 Strategic Reasoning Checkpoints

Every problem features interactive stepwise discovery: propose mathematical ideas, explore strategic checkpoints, and verify governing invariants.
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Showing 1–24 of 4,079 problems
Page 1 of 170
algebraolympiad
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Nested Iteration Functional Equation on Nonnegative Integers

Let Z⩾0\mathbb{Z}_{\geqslant 0} be the set of all nonnegative integers. Find all functions f:Z⩾0→Z⩾0f: \mathbb{Z}_{\geqslant 0} \rightarrow \mathbb{Z}_{\geqslant 0} satisfying f(f(f(n)))=f(n+1)+1 f(f(f(n))) = f(n+1) + 1 for all n∈Z⩾0n \in \mathbb{Z}_{\geqslant 0}.
#Olympiad#Grade 1 Masterpiece#algebra
ID: DIV-G1-001
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number theorymedium
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Maximal-Sum Squeeze Forcing the Identity Sequence

Determine all integers x1,x2,⋯ ,x9,x10x_1, x_2, \cdots, x_9, x_{10} such that 0<x1<x2<⋯<x9<x10andx9 x10⩽2(x1+x2+⋯+x9).0 < x_1 < x_2 < \cdots < x_9 < x_{10} \quad \text{and} \quad x_9\, x_{10} \leqslant 2\left(x_1 + x_2 + \cdots + x_9\right).
#Olympiad#Grade 1 Masterpiece#number_theory
ID: DIV-G1-002
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number theoryolympiad
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Triangular Numbers Around the Circle: When Does Every Child Get Candy?

During a break, nn children sit in a circle around their teacher. The teacher selects one child and gives him a candy, then skips the next child and gives a candy to the next one, then skips 22 and gives a candy to the next one, then skips 33, and so on. Determine all values of nn for which eventually, perhaps after many rounds, every child has received at least one candy.
#Olympiad#Grade 1 Masterpiece#number_theory
ID: DIV-G1-003
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number theoryolympiad
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Order Periodicity Transport Forces Q Constant

Let PP and QQ be polynomials with integer coefficients such that any polynomial with rational coefficients dividing both PP and QQ is constant. Furthermore, assume that for all n∈N∗n \in \mathbb{N}^{*}, P(n)P(n) and Q(n)Q(n) are strictly positive, and 2Q(n)−12^{Q(n)}-1 divides 3P(n)−13^{P(n)}-1. Prove that QQ is constant.
#Olympiad#Grade 1 Masterpiece#number_theory
ID: DIV-G1-004
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algebraolympiad
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Nested Iteration Functional Equation: Injectivity, Range Defect, and Modular Classification

Let Z⩾0\mathbb{Z}_{\geqslant 0} be the set of all nonnegative integers. Find all functions f:Z⩾0→Z⩾0f:\mathbb{Z}_{\geqslant 0}\to\mathbb{Z}_{\geqslant 0} satisfying f(f(f(n)))=f(n+1)+1f(f(f(n)))=f(n+1)+1 for all n∈Z⩾0n\in\mathbb{Z}_{\geqslant 0}. Here fkf^{k} denotes the kk-th iterate of ff, with f0=idf^{0}=\mathrm{id}.
#Olympiad#Grade 1 Masterpiece#algebra
ID: DIV-G1-005
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algebraolympiad
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Nested Iteration Functional Equation: Unit Shifts and Mod-4 Orbit Collapse

Find all functions f:N→Nf:\mathbb{N}\to\mathbb{N} such that for every n∈Nn\in\mathbb{N},

f(f(f(n)))=f(n+1)+1.f(f(f(n)))=f(n+1)+1.

(Here N={0,1,2,… }\mathbb{N}=\{0,1,2,\dots\}.)

#Olympiad#Grade 1 Masterpiece#algebra
ID: DIV-G1-006
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number theoryolympiad
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Self-Compositional Collapse on Zp\mathbb{Z}_p via Mersenne Divisors

Let p>2p > 2 be a fixed prime number. Find all functions f:Z→Zpf: \mathbb{Z} \to \mathbb{Z}_p, where Zp={0,1,…,p−1}\mathbb{Z}_p = \{0, 1, \ldots, p-1\}, such that p∣f(f(n))−f(n+1)+1p \mid f(f(n)) - f(n+1) + 1 and f(n+p)=f(n)f(n+p) = f(n) for all integers nn.
#Olympiad#Grade 1 Masterpiece#number_theory
ID: DIV-G1-007
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combinatorics probabilityolympiad
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Balanced One-Way Orientations of Even Road Networks

A country has finitely many cities and finitely many roads. Each road joins two different cities, and each pair of cities is joined by at most one road. Every road is traversable in both directions, and the network is connected: every city can be reached from every other city, possibly via detours. Moreover, an even number of roads leads away from each city. The government now converts every road into a one-way street so that, at each city, the number of roads leading out equals the number of roads leading in. (a) Show that this is always possible. (b) Show that no matter how the government implements its plan, one can still reach any city from any other.
#Olympiad#Grade 1 Masterpiece#combinatoricprobability
ID: DIV-G1-008
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combinatorics probabilityolympiad
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Exceptional Shift Constants Minimizing Periodic Functional Solutions

Find all integers c∈{0,1,…,2016}c \in \{0,1,\ldots,2016\} for which the number of functions f:Z→{0,1,…,2016}f:\mathbb{Z}\to\{0,1,\ldots,2016\} satisfying (1) ff is periodic with period 20172017, and (2) f(f(x)+f(y)+1)−f(f(x)+f(y))≡c(mod2017)f(f(x)+f(y)+1)-f(f(x)+f(y))\equiv c\pmod{2017} for all integers x,yx,y, is minimal.
#Olympiad#Grade 1 Masterpiece#combinatoricprobability
ID: DIV-G1-009
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combinatorics probabilityolympiad
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Parity Trap: The Odd-Cycle Avoidance Game

David and Jacob alternately take turns; on each turn a player chooses two of the n≥3n \geq 3 given points (no three collinear) and connects them by a new line segment. The first player to complete a cycle consisting of an odd number of segments loses, where every endpoint in the cycle must be among the original nn points (intersections of segments do not count as vertices). Assuming David moves first, determine all nn for which David has a winning strategy.
#Olympiad#Grade 1 Masterpiece#combinatoricprobability
ID: DIV-G1-010
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geometryolympiad
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Parallel Chords, Reflection Across a Chord, and Equal Distances to Lines

Let A,B,C,D,E,FA, B, C, D, E, F be points on a circle with AE∥BDAE \parallel BD and BC∥DFBC \parallel DF. Reflecting the point DD across the line CECE produces the point XX. Show that the distance from XX to the line EFEF equals the distance from BB to the line ACAC: dist⁡(X,EF)=dist⁡(B,AC)\operatorname{dist}(X, EF) = \operatorname{dist}(B, AC).
#Olympiad#Grade 1 Masterpiece#geometry
ID: DIV-G1-011
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number theoryolympiad
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Unique Cyclic Placement of Primes via the Special-Pair Lemma

Let kk be a positive integer and let SS be a finite set of odd prime numbers. Prove that there is at most one way (modulo rotation and reflection) to place the elements of SS around a circle such that the product of any two neighbors is of the form x2+x+kx^{2}+x+k for some positive integer xx.
#Olympiad#Grade 1 Masterpiece#number_theory
ID: DIV-G1-012
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number theoryolympiad
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Unique Cyclic Placement of Primes via Quadratic Neighbor Products

Let kk be a positive integer and let SS be a finite set of odd prime numbers. Prove that there is at most one way (modulo rotation and reflection) to place the elements of SS around a circle such that the product of any two neighbors is of the form x2+x+kx^{2}+x+k for some positive integer xx.
#Olympiad#Grade 1 Masterpiece#number_theory
ID: DIV-G1-013
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mixed otherolympiad
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Involution Extraction in a Nested Functional Equation

#Olympiad#Grade 1 Masterpiece#mixed_other
ID: DIV-G1-014
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geometryolympiad
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Excircle Contact Triangle: Inversion Symmetry of Orthic Lines

Given an acute triangle △ABC\triangle ABC with ACneqABAC neq AB and circumcircle (C)(C), let (C1)(C_1) be the AA-excircle with center IaI_a, tangent to BCBC at DD and to the extensions of AB,ACAB, AC at E,ZE, Z. Let I,LI, L be the intersections of (C)(C) and (C1)(C_1), HH the orthocenter of △EDZ\triangle EDZ, and NN the midpoint of EZEZ. The parallel through IaI_a to HLHL meets HIHI at GG. Prove that the perpendicular (e)(e) through NN to BCBC and the parallel (δ)(\delta) through GG to ILIL meet each other on the line HIaHI_a.
#Olympiad#Grade 1 Masterpiece#geometry
ID: DIV-G1-015
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combinatorics probabilityolympiad
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Thirteen Lattice Points Force a Lattice Centroid

A lattice point in the plane is a point whose coordinates are both integers. The centroid of four points (xi,yi)(x_i, y_i), i=1,2,3,4i=1,2,3,4, is the point (x1+x2+x3+x44,y1+y2+y3+y44)\left(\frac{x_1+x_2+x_3+x_4}{4}, \frac{y_1+y_2+y_3+y_4}{4}\right). Let nn be the largest natural number with the following property: There are nn distinct lattice points in the plane such that the centroid of any four of them is not a lattice point. Prove that n=12n = 12.
#Olympiad#Grade 1 Masterpiece#combinatoricprobability
ID: DIV-G1-016
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algebraolympiad
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Parity-Flipping Involutions Tamed by an Odd-Sum Inequality

Find all functions f:Z→Zf: \mathbb{Z} \to \mathbb{Z} such that (i) f(f(x))=xf(f(x)) = x for all integers xx, and (ii) f(x)+f(y)≥x+yf(x) + f(y) \ge x + y for all integers x,yx, y with x+yx + y odd.
#Olympiad#Grade 1 Masterpiece#algebra
ID: DIV-G1-017
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number theoryolympiad
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Signed Subset Sums and the Pigeonhole Threshold Modulo 2013

Let n>0n > 0 be an integer. Anne writes nn distinct positive integers on the board. Bernard then erases some of these numbers (possibly none, but not all). In front of each remaining number he writes a ++ or a −-, and performs the corresponding addition. If the result is divisible by 20132013, Bernard wins; otherwise, Anne wins. Determine, according to the value of nn, which player has a winning strategy.
#Olympiad#Grade 1 Masterpiece#number_theory
ID: DIV-G1-018
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number theoryolympiad
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Cyclic Digit Balance for Multiples of 2023

Let n⩾1n \geqslant 1 be an integer. Morgane writes on the board, in base 10, the numbers 2023,2023×2,…,2023×n2023, 2023 \times 2, \ldots, 2023 \times n. For every digit cc between 11 and 99, she notes dc(n)\mathrm{d}_{c}(n), the number of occurrences of the digit cc on the board. Prove that there are infinitely many integers n⩾1n \geqslant 1 for which the nine numbers d1(n),…,d9(n)\mathrm{d}_{1}(n), \ldots, \mathrm{d}_{9}(n) take exactly two values.
#Olympiad#Grade 1 Masterpiece#number_theory
ID: DIV-G1-019
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combinatorics probabilityolympiad
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Fibonacci Cycles Cap the Palette of a Power-of-Two Table

The rows and columns of a 2n×2n2^{n}\times 2^{n} table are numbered from 00 to 2n−12^{n}-1. The cells are coloured so that for every 0≤i,j≤2n−10\le i,j\le 2^{n}-1, the jj-th cell of row ii — namely cell (i,j)(i,j) — has the same colour as the jj-th cell of column i+j(mod2n)i+j \pmod{2^{n}}, namely cell (j, i+j mod 2n)(j,\, i+j \bmod 2^{n}). Prove that the maximal possible number of colours is 2n2^{n}.
#Olympiad#Grade 1 Masterpiece#combinatoricprobability
ID: DIV-G1-020
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algebraolympiad
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Odd-Valued Reflection Equation and Quasi-Periodic Classification on Z\mathbb{Z}

Let 2Z+12\mathbb{Z}+1 denote the set of odd integers. Find all functions f:Z→2Z+1f:\mathbb{Z}\to 2\mathbb{Z}+1 satisfying f(x+f(x)+y)+f(x−f(x)−y)=f(x+y)+f(x−y)f(x+f(x)+y)+f(x-f(x)-y)=f(x+y)+f(x-y) for every x,y∈Zx,y\in\mathbb{Z}.
#Olympiad#Grade 1 Masterpiece#algebra
ID: DIV-G1-021
Solve Problem
number theoryolympiad
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Orbit Decomposition and LCM Periodicity of f(n)−nf(n)-n

Let Z>0\mathbb{Z}_{>0} denote the set of positive integers and let f:Z>0→Z>0f:\mathbb{Z}_{>0}\to\mathbb{Z}_{>0} satisfy: (i) for all m,n∈Z>0m,n\in\mathbb{Z}_{>0}, fn(m)−mn∈Z>0\frac{f^{n}(m)-m}{n}\in\mathbb{Z}_{>0}; (ii) the set Z>0∖{f(n):n∈Z>0}\mathbb{Z}_{>0}\setminus\{f(n):n\in\mathbb{Z}_{>0}\} is finite. Prove that the sequence f(1)−1, f(2)−2, f(3)−3,…f(1)-1,\ f(2)-2,\ f(3)-3,\ldots is periodic.
#Olympiad#Grade 1 Masterpiece#number_theory
ID: DIV-G1-022
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algebraolympiad
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Quasi-Periodic Descent for an Odd-Valued Functional Equation on Z\mathbb{Z}

Let 2Z+12\mathbb{Z}+1 denote the set of odd integers. Find all functions f:Z→2Z+1f:\mathbb{Z}\to 2\mathbb{Z}+1 satisfying f(x+f(x)+y)+f(x−f(x)−y)=f(x+y)+f(x−y)f(x+f(x)+y)+f(x-f(x)-y)=f(x+y)+f(x-y) for every x,y∈Zx,y\in\mathbb{Z}. (Answer: fix an odd positive integer dd, an integer kk, and odd integers ℓ0,ℓ1,…,ℓd−1\ell_0,\ell_1,\dots,\ell_{d-1}; then f(md+i)=2kmd+ℓidf(md+i)=2kmd+\ell_i d for m∈Zm\in\mathbb{Z}, i=0,…,d−1i=0,\dots,d-1, and these are all solutions.)
#Olympiad#Grade 1 Masterpiece#algebra
ID: DIV-G1-023
Solve Problem
algebraolympiad
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Orbit Dichotomy in an Iterated Functional Equation

Determine all functions f:Z→Zf:\mathbb{Z}\to\mathbb{Z} such that fa2+b2(a+b)=a f(a)+b f(b)f^{a^{2}+b^{2}}(a+b)=a\,f(a)+b\,f(b) for every a,b∈Za,b\in\mathbb{Z}, where fnf^{n} denotes the nn-th iterate of ff, i.e., f0(x)=xf^{0}(x)=x and fn+1(x)=f(fn(x))f^{n+1}(x)=f\left(f^{n}(x)\right) for all n⩾0n\geqslant 0.
#Olympiad#Grade 1 Masterpiece#algebra
ID: DIV-G1-024
Solve Problem
Page 1 of 170