TSTST #0jee — TSTST · USAMO Guide - Olympiad Mathematics Learning Platform

TSTST #0jee

TSTSTHard

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Understand the scale

Scale 0.5: Easiest math competition problems, often solvable even for early middle school students without experience. 1: Problems strictly for beginners, often at a moderate middle school level. 1.5: Problems for stronger beginner students, on the level of the middling problems in most middle school contests. 2: For motivated beginners, harder questions from the previous categories. 2.5: More advanced beginner problems, hardest questions from previous categories. 3: Early intermediate problems that require more creative thinking. 3.5: Tougher early intermediate problems that consistently stretch into higher-level creative thinking or conceptual knowledge. 4: Intermediate-level problems. 4.5: Upper intermediate problems approaching the upper bound for non-invitational sprint math competitions. 5: More difficult AIME problems or simple proof-based Olympiad-style problems. 6: High-level AIME-styled questions or introductory Olympiad-level questions. 7: Tougher Olympiad-level questions, may require more technical knowledge. 8: High-level Olympiad-level questions. 9: Expert Olympiad-level questions. 9.5: The hardest problems appearing on Olympiads which the strongest students could reasonably solve. 10: Historically notorious problems, generally unsuitable for even very hard competitions because they are exceedingly tedious, long, and difficult.

Problem

Let $ABCD$ be a quadrilateral with $AC = BD$ . Diagonals $AC$ and $BD$ meet at $P$ . Let $\omega_1$ and $O_1$ denote the circumcircle and circumcenter of triangle $ABP$ . Let $\omega_2$ and $O_2$ denote the circumcircle and circumcenter of triangle $CDP$ . Segment $BC$ meets $\omega_1$ and $\omega_2$ again at $S$ and $T$ (other than $B$ and $C$ ), respectively. Let $M$ and $N$ be the midpoints of minor arcs $\widehat{SP}$ (not including $B$ ) and $\widehat{TP}$ (not including $C$ ). Prove that $MN \parallel O_1O_2$ .

(This problem was suggested by Steve Dinh.)