TSTST #0jga

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.

Triangle centers: centroid, incenter, circumcenter, orthocenter, Euler line, nine-point circleCyclic quadrilateralsAngle chasing

Problem

In scalene triangle $ABC$ , let the feet of the perpendiculars from $A$ to $BC$ , $B$ to $CA$ , $C$ to $AB$ be $A_1$ , $B_1$ , $C_1$ , respectively. Denote by $A_2$ the intersection of lines $BC$ and $B_1C_1$ . Define $B_2$ and $C_2$ analogously. Let $D$ , $E$ , $F$ be the respective midpoints of sides $BC$ , $CA$ , $AB$ . Show that the perpendiculars from $D$ to $AA_2$ , $E$ to $BB_2$ , and $F$ to $CC_2$ are concurrent. (This problem was suggested by Ian Le.)

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