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.