There are $m$ foxes and $n$ bunnies in $m+n$ seats around a circular table, all distinguishable. A neighboring fox and bunny may swap positions. From a fixed starting configuration, determine the number of configurations reachable by a sequence of swaps. Rotations and reflections are considered distinct.
Proposed by Milan Haiman.
Manual essay; 1 editorial point, not official contest scoring.
Let $ABC$ be acute and scalene, with no angle $60^\circ$, and circumcircle $\omega$. Let $\Delta_B$ be the equilateral triangle inscribed in $\omega$ with one vertex $B$, and let $\ell_B$ pass through its other two vertices. Define $\Delta_C,\ell_C$ analogously. Let $Y=AC\cap\ell_B$ and $Z=AB\cap\ell_C$. The circumcircle of $AYZ$ meets $\omega$ again at $P\ne A$. Let $D=BC\cap YZ$ and $E=PA\cap YZ$. Prove $PE=PD$.
Proposed by Rogelio Guerrero.
Manual essay; 1 editorial point, not official contest scoring.
Triangle $ABC$ has circumcircle $\omega$ and circumcenter $O$. Lines $AO$ and $BC$ meet at $D$. Let $X$ be the $A$-excenter of triangle $ABD$, and $Y$ the $A$-excenter of triangle $ACD$. Prove that if $X\in\omega$, then $Y\in\omega$.
Proposed by Carl Schildkraut.
Manual essay; 1 editorial point, not official contest scoring.
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.
Emily has a red sheet of paper. She draws $2026$ circles, not necessarily of equal size, colors one circle black, and cuts around all $2026$ circumferences. She separates the paper into at least $2$ black pieces and some red pieces. Is it possible that all the black pieces are congruent?
Proposed by Carl Schildkraut.
Manual essay; 1 editorial point, not official contest scoring.
Fix positive integers $k$ and $d$.
Prove that for all sufficiently large odd positive integers $n$,
the digits of the base-$2n$ representation of $n^k$ are all greater than $d$.
Proposed by John Berman.
Manual essay; 1 editorial point, not official contest scoring. Automatic answer fields are intentionally empty.
Let $m,n$ be positive integers and let $\mathcal R$ be a $2m\times2n$ grid of unit squares. A domino is a $1\times2$ or $2\times1$ rectangle. An up-right path goes from the lower-left corner to the upper-right corner using exactly $2m+2n$ grid edges. Find the number of these paths dividing $\mathcal R$ into two possibly empty subsets, each tileable with dominoes.
Proposed by Wilbert Chu.
Manual essay; 1 editorial point, not official contest scoring.
Let $n$ be a positive integer and let $a_0\ge a_1\ge\cdots\ge a_n\ge0$ be integers. Prove that $$\sum_{i=0}^n i\binom{a_i}{2}\le\frac12\binom{a_0+a_1+\cdots+a_n}{2}.$$
Proposed by Hung-Hsun Yu.
Manual essay; 1 editorial point, not official contest scoring.
Let $H$ be the orthocenter of an acute triangle $ABC$,
let $F$ be the foot of the altitude from $C$ to $AB$,
and let $P$ be the reflection of $H$ across $BC$.
Suppose that the circumcircle of triangle $AFP$ intersects line $BC$
at two distinct points $X$ and $Y$.
Prove that $CX = CY$.
Proposed by Carl Schildkraut.
Manual essay; 1 editorial point, not official contest scoring. Automatic answer fields are intentionally empty.
Let $S$ be a set of integers such that (i) $\{1,2,\ldots,2025\}\subseteq S$; (ii) if $a,b\in S$ and $\gcd(a,b)=1$, then $ab\in S$; (iii) if $s\in S$ and $s+1$ is composite, then all positive divisors of $s+1$ belong to $S$. Prove that $S$ contains all positive integers.
Proposed by Enrique Treviño.
Manual essay; 1 editorial point, not official contest scoring.
Let $ABCD$ be cyclic, with $AB=7$ and $CD=8$. Points $P,Q$ lie on segment $AB$ with $AP=BQ=3$. Points $R,S$ lie on segment $CD$ with $CR=DS=2$. Prove that $PQRS$ is cyclic.
Proposed by Evan o’Dorney. Source: Evan Chen’s third-party compilation; https://aops.com/community/p30216434. One editorial point; manual grading.
Let $m,n$ be positive integers and let $S$ be the lattice points $(x,y)$ with $1\le x\le2m$ and $1\le y\le2n$. A configuration of $mn$ axis-parallel rectangles is happy if every point of $S$ is a vertex of exactly one rectangle. Prove that the number of happy configurations is odd.
Proposed by Serena An and Claire Zhang. Source: Evan Chen’s third-party compilation; https://aops.com/community/p30216444. One editorial point; manual grading.
Let $n\ge3$. Rowan may permute rows and Colin may permute columns of an $n\times n$ grid colored red or blue. A coloring is orderly if every row permutation can be undone by a column permutation, and every column permutation can be undone by a row permutation, restoring the original coloring in each case. How many orderly colorings are there?
Proposed by Alec Sun. Source: Evan Chen’s third-party compilation; https://aops.com/community/p30227193. One editorial point; manual grading.
Point $D$ is selected inside acute triangle $ABC$
so that $\angle DAC = \angle ACB$ and $\angle BDC = 90^\circ + \angle BAC$.
Point $E$ is chosen on ray $BD$ so that $AE = EC$.
Let $M$ be the midpoint of $BC$.
Show that line $AB$ is tangent to the circumcircle of triangle $BEM$.
Proposed by Anton Trygub. One nonofficial editorial point; manual essay grading.
In an acute triangle $ABC$, let $M$ be the midpoint of $\overline{BC}$.
Let $P$ be the foot of the perpendicular from $C$ to $AM$.
Suppose that the circumcircle of triangle $ABP$
intersects line $BC$ at two distinct points $B$ and $Q$.
Let $N$ be the midpoint of $\overline{AQ}$.
Prove that $NB = NC$.
Proposed by Holden Mui. One nonofficial editorial point; manual essay grading.