HMMT 2013 #0jfc — HMMT 2013 · USAMO Guide - Olympiad Mathematics Learning Platform

HMMT 2013 #0jfc

HMMT 2013Hard

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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

Problem: Let $A_{1} A_{2} A_{3} A_{4} A_{5} A_{6}$ be a convex hexagon such that $A_{i} A_{i+2} \parallel A_{i+3} A_{i+5}$ for $i=1,2,3$ (we take $A_{i+6}=A_{i}$ for each $i$ ). Segment $A_{i} A_{i+2}$ intersects segment $A_{i+1} A_{i+3}$ at $B_{i}$ , for $1 \leq i \leq 6$ , as shown. Furthermore, suppose that $\triangle A_{1} A_{3} A_{5} \cong \triangle A_{4} A_{6} A_{2}$ . Given that $[A_{1} B_{5} B_{6}]=1$ , $[A_{2} B_{6} B_{1}]=4$ , and $[A_{3} B_{1} B_{2}]=9$ (by $[X Y Z]$ we mean the area of $\triangle X Y Z$ ), determine the area of hexagon $B_{1} B_{2} B_{3} B_{4} B_{5} B_{6}$ .