MathGoldMedalist
Sign InRegister

USAMO 2026

Problems, Step-by-Step Hints, and Solutions

Prepare for USAMO →

← USAMO

1.
Let nn be an integer greater than 11. For which real numbers xx is
⌊nx⌋−∑k=1n⌊kx⌋k\lfloor nx\rfloor-\sum_{k=1}^{n}\frac{\lfloor kx\rfloor}{k}
maximal, and what is the maximal value that this expression can take? Note: ⌊z⌋\lfloor z\rfloor denotes the greatest integer less than or equal to zz.
2.
Annie is playing a game where she starts with a row of positive integers, written on a blackboard, each of which is a power of 2. On each turn, she can erase two adjacent numbers and replace them with a power of 2 that is greater than either of the erased numbers. This shortens the row of numbers, and she continues to take turns until only one number remains. Annie wins the game if the final remaining number is less than 4 times the sum of the original numbers. Is it always possible for Annie to win, regardless of the starting row of numbers?
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 two vertices of ΔB\Delta_B other than BB. Let ΔC\Delta_C and ℓC\ell_C be defined analogously. Let YY be the intersection of ACAC and ℓB\ell_B, and let ZZ be the intersection of ABAB and ℓC\ell_C.

Let NN be the midpoint of minor arc BCBC on ω\omega. Let R\mathcal R be the triangle formed by ℓB\ell_B, ℓC\ell_C, and the tangent to ω\omega through NN. Prove that the circumcircle of AYZAYZ and the incircle of R\mathcal R are tangent.
4.
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}.
5.
Let ABCABC be a triangle. Points DD, EE, and FF lie on sides BCBC, CACA, and ABAB, respectively, such that
∠AFE=∠BDF=∠CED.\angle AFE=\angle BDF=\angle CED.
Let OAO_A, OBO_B, and OCO_C be the circumcenters of triangles AFEAFE, BDFBDF, and CEDCED, respectively. Let MM, NN, and OO be the circumcenters of triangles ABCABC, DEFDEF, and OAOBOCO_AO_BO_C, respectively. Prove that OM=ONOM=ON.
6.
Let aa and bb be positive integers such that φ(ab+1)\varphi(ab+1) divides a2+b2+1a^2+b^2+1. Prove that aa and bb are Fibonacci numbers.
Note: Euler's totient function φ(n)\varphi(n) is the number of positive integers less than or equal to nn that are relatively prime to nn.