HMIC 2023 #0kvw

HMIC 2023Hard

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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.

Triangle centers: centroid, incenter, circumcenter, orthocenter, Euler line, nine-point circleTangentsInversionCartesian coordinatesAngle chasing

Problem

Problem:

Triangle $A B C$ has incircle $\omega$ and $A$ -excircle $\omega_{A}$ . Circle $\gamma_{B}$ passes through $B$ and is externally tangent to $\omega$ and $\omega_{A}$ . Circle $\gamma_{C}$ passes through $C$ and is externally tangent to $\omega$ and $\omega_{A}$ . If $\gamma_{B}$ intersects line $B C$ again at $D$ , and $\gamma_{C}$ intersects line $B C$ again at $E$ , prove that $B D=E C$ .

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