MathGoldMedalist
Sign InRegister

USAJMO 2026

Problems, Step-by-Step Hints, and Solutions

Prepare for USAJMO →

← USAJMO

1.
Let a,b,ca,b,c be distinct positive integers such that ab+c=c2ab+c=c^2. Prove that (a−b)2≥4c(a-b)^2\ge 4c.
2.
There are mm foxes and nn rabbits sitting in m+nm+n seats around a circular table, where all animals are distinguishable. If a fox and a rabbit are sitting next to each other, they may swap positions with each other. For a fixed starting configuration, determine the number of configurations that can be reached by a sequence of swaps.

Note: Rotations and reflections of a configuration are considered distinct.
3.
Let ABCABC be an acute scalene triangle with no angle equal to 60∘60^\circ. Let ω\omega be the circumcircle of ABCABC. Let ΔB\Delta_B be the equilateral triangle with three vertices on ω\omega, one of which is BB. Let ℓB\ell_B be the line through the other two vertices of ΔB\Delta_B. Define ΔC\Delta_C and ℓC\ell_C analogously. Let Y=AC∩ℓBY=AC\cap\ell_B and Z=AB∩ℓCZ=AB\cap\ell_C. Suppose the circumcircle of AYZAYZ intersects ω\omega again at P≠AP\ne A. Let D=BC∩YZD=BC\cap YZ and E=PA∩YZE=PA\cap YZ. Prove that PE=PDPE=PD.
4.
Triangle ABCABC has circumcircle ω\omega and circumcenter OO. Lines AOAO and BCBC meet at point DD. Let XX be the AA-excenter of △ABD\triangle ABD and let YY be the AA-excenter of △ACD\triangle ACD. Prove that if XX lies on ω\omega, then YY also lies on ω\omega.
5.
A positive integer nn is called solitary if, for any nonnegative integers aa and bb such that a+b=na+b=n, either aa or bb contains the digit 11 when written in base 1010. Determine, with proof, the number of solitary integers less than 10202610^{2026}.
6.
Emily has a red sheet of paper. She draws 2026 circles (not necessarily of equal size) on the piece of paper. She chooses a circle to color black, then cuts the paper around the circumference of all 2026 circles. She then separates the pieces of paper into at least 2 black pieces and some number of red pieces. Is it possible that all black pieces are congruent?