USEMO 2022

A stick is defined as a $1 \times k$ or $k\times 1$ rectangle for any integer $k \ge 1$ . We wish to partition the cells of a $2022 \times 2022$ chessboard into $m$ non-overlapping sticks, such that any two of these $m$ sticks share at most one unit of perimeter. Determine the smallest $m$ for which this is possible.

Holden Mui

A function $\psi \colon {\mathbb Z} \to {\mathbb Z}$ is said to be zero-requiem if for any positive integer $n$ and any integers $a_1$ , $\ldots$ , $a_n$ (not necessarily distinct), the sums $a_1 + a_2 + \dots + a_n$ and $\psi(a_1) + \psi(a_2) + \dots + \psi(a_n)$ are not both zero.

Let $f$ and $g$ be two zero-requiem functions for which $f \circ g$ and $g \circ f$ are both the identity function (that is, $f$ and $g$ are mutually inverse bijections). Given that $f+g$ is not a zero-requiem function, prove that $f \circ f$ and $g \circ g$ are both zero-requiem.

Sutanay Bhattacharya

Point $P$ lies in the interior of a triangle $ABC$ . Lines $AP$ , $BP$ , and $CP$ meet the opposite sides of triangle $ABC$ at $A$ ', $B'$ , and $C'$ respectively. Let $P_A$ the midpoint of the segment joining the incenters of triangles $BPC'$ and $CPB'$ , and define points $P_B$ and $P_C$ analogously. Show that if $$AB'+BC'+CA'=AC'+BA'+CB'$$ then points $P,P_A,P_B,$ and $P_C$ are concyclic.

Nikolai Beluhov