Question bank USAMO 2026

Fix an integer $n \ge 2$.
For which real numbers $x$ is

$$\lfloor nx\rfloor-\sum_{k=1}^n\frac{\lfloor kx\rfloor}{k}$$

maximal, and what is the maximal value that this expression can take?

Proposed by Titu Andreescu and Gabriel Dospinescu.

Manual essay; 1 editorial point, not official contest scoring. Automatic answer fields are intentionally empty.

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?

Proposed by Andrew Dittmer.

Manual essay; 1 editorial point, not official contest scoring. Automatic answer fields are intentionally empty.

Let $ABC$ be an acute scalene triangle with no angle equal to $60^\circ$.
Let $\omega$ be the circumcircle of $ABC$.
Let $\Delta_B$ be the equilateral triangle with three vertices on $\omega$,
one of which is $B$.
Let $\ell_B$ be the line through the two vertices of $\Delta_B$ other than $B$.
Let $\Delta_C$ and $\ell_C$ be defined analogously.
Let $Y$ be the intersection of $AC$ and $\ell_B$,
and let $Z$ be the intersection of $AB$ and $\ell_C$.

Let $N$ be the midpoint of minor arc $BC$ on $\omega$.
Let $\mathcal{R}$ be the triangle formed by $\ell_B$, $\ell_C$, and the tangent to $\omega$ through $N$.
Prove that the circumcircle of $AYZ$ and the incircle of $\mathcal{R}$ are tangent.

Proposed by Rogelio Guerrero.

Manual essay; 1 editorial point, not official contest scoring. Automatic answer fields are intentionally empty.

A positive integer $n$ is called solitary if,
for any non-negative integers $a$ and $b$ such that $a+b=n$,
either $a$ or $b$ contains the digit $1$.
Determine, with proof, the number of solitary integers less than $10^{2026}$.

Proposed by Carl Schildkraut.

Manual essay; 1 editorial point, not official contest scoring. Automatic answer fields are intentionally empty.

Let $ABC$ be a triangle.
Points $D$, $E$, and $F$ lie on sides $BC$, $CA$, and $AB$, respectively, such that

$$\angle AFE = \angle BDF = \angle CED.$$

Let $O_A$, $O_B$, and $O_C$ be the circumcenters of triangles $AFE$, $BDF$, and $CED$, respectively.
Let $M$, $N$, and $O$ be the circumcenters of triangles $ABC$, $DEF$, and $O_AO_BO_C$, respectively.
Prove that $OM = ON$.

Proposed by Serena An.

Manual essay; 1 editorial point, not official contest scoring. Automatic answer fields are intentionally empty.

Let $a$ and $b$ be positive integers such that
$\varphi(ab+1)$ divides $a^2+b^2+1$.
Prove that $a$ and $b$ are Fibonacci numbers.

Proposed by Raymond Feng and Eric Shen.

Manual essay; 1 editorial point, not official contest scoring. Automatic answer fields are intentionally empty.