HMMT November 2014 #0jk4

HMMT November 2014Hard

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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.

Cyclic quadrilateralsAngle chasingTriangle trigonometry

Problem

Problem:

Let $ABC$ be a triangle with $AB = 13$ , $BC = 14$ , and $CA = 15$ . We construct isosceles right triangle $ACD$ with $\angle ADC = 90^{\circ}$ , where $D$ , $B$ are on the same side of line $AC$ , and let lines $AD$ and $CB$ meet at $F$ . Similarly, we construct isosceles right triangle $BCE$ with $\angle BEC = 90^{\circ}$ , where $E$ , $A$ are on the same side of line $BC$ , and let lines $BE$ and $CA$ meet at $G$ . Find $\cos \angle AGF$ .

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