Focos Kernel v1 · Adaptive Capability Engine
Find out what your technical mathematical mind actually looks like.
Focos maps what you can do across Olympiad Mathematics — Algebra, Combinatorics, Geometry, and Number Theory — then trains the gap. Not a course. A model of you.
Olympiad Capability Map PreviewIllustrative Example
Algebra & Inequalities
Extremal BoundsSOLID (VERIFIED)
Cauchy-Schwarz InvariantsSTRIPED (PRESUMED)
Combinatorics & Discrete
Discrete MonovariantsSOLID (VERIFIED)
Pigeonhole PartitionsFADED (STALE)
Number Theory
p-adic ValuationsSOLID (VERIFIED)
Descent on Diophantine— (UNKNOWN)
Proof Zoom Flagship Demo · IMO 1988 Problem 6
Same problem. Same proof. Three completely different conversations.
The engine decides what to ask you based on your latent capability model — and explains why.
Replay Step: 4 of 6
Focos Prompt
If a, b are positive integers such that (a² + b²)/(ab + 1) is an integer k, show k is a perfect square.
Elena
Vieta jump, minimising a + b.
Focos (Compiled)
✓ Taking a minimal pair with a ≥ b, reading the equation as x² − (kb)x + (b²−k) = 0 with root a, the second root x₂ = kb − a is an integer. Casework on sign leaves x₂ = 0, hence k = b².
Focos Prompt
One thing that isn't closed: in the case x₂ > 0 you get another solution — why is it smaller in a+b?