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.
Let $n$ be an odd positive integer. A maximal grid-aligned configuration $C$ consists of $(n^2-1)/2$ identical dominoes on an $n\times n$ board of unit squares. Each covers two neighboring squares, with no overlap, so one square is uncovered. A move slides, without rotation, a domino to cover the uncovered square and leave another square uncovered. Let $k(C)$ be the number of distinct configurations obtainable by any number of such moves, including zero. Find the maximum of $k(C)$ in terms of $n$.
Proposed by Holden Mui. Source: Evan Chen’s third-party compilation; https://aops.com/community/p27349423. One editorial point; manual grading.
Blake and Ruby play on an infinite grid, initially white. Starting with Blake, they alternate turns. Blake colors one white square blue; Ruby colors two white squares red. Blake may end the game. His score is the number of unit squares in the largest simple polygon consisting only of blue unit squares. What is the largest score Blake can guarantee?
Proposed by David Torres. Source: Evan Chen’s third-party compilation; https://aops.com/community/p27349414. One editorial point; manual grading.
Positive integers $a$ and $N$ are fixed,
and $N$ positive integers are written on a blackboard.
Alice and Bob play the following game.
On Alice's turn, she must replace some integer $n$ on the board with $n+a$,
and on Bob's turn he must replace some even integer $n$ on the board with $n/2$.
Alice goes first and they alternate turns.
If on his turn Bob has no valid moves, the game ends.
After analyzing the $N$ integers on the board, Bob realizes that,
regardless of what moves Alice makes,
he will be able to force the game to end eventually.
Show that, in fact, for this value of $a$ and these $N$ integers on the board,
the game is guaranteed to end regardless of Alice's or Bob's moves.
Proposed by Carl Schildkraut. One nonofficial editorial point; manual essay grading. Source reference: https://aops.com/community/p27349336.
An isosceles triangle $ABC$, with $AB=AC$, is inscribed in a circle $\omega$. A point $D$ lies in the interior of segment $BC$, with $BD\ne DC$. Ray $AD$ meets $\omega$ again at $E$. A point $F\ne E$ on $\omega$ satisfies $\angle DFE=90^\circ$. Line $FE$ meets rays $AB,AC$ at $X,Y$, respectively. Prove $\angle XDE=\angle EDY$.
Proposed by Anton Trygub. Source: Evan Chen’s third-party compilation; https://aops.com/community/p27349508. One editorial point; manual grading.