HMMT Spring 2021 Guts Round #0knd

HMMT Spring 2021 Guts RoundHard

Community difficultyNo ratings yetBe the first to rate this problem.

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 trigonometryAngle chasing

Problem

Problem:

Let $AD$ , $BE$ , and $CF$ be segments sharing a common midpoint, with $AB < AE$ and $BC < BF$ . Suppose that each pair of segments forms a $60^{\circ}$ angle, and that $AD = 7$ , $BE = 10$ , and $CF = 18$ . Let $K$ denote the sum of the areas of the six triangles $\triangle ABC$ , $\triangle BCD$ , $\triangle CDE$ , $\triangle DEF$ , $\triangle EFA$ , and $\triangle FAB$ . Compute $K \sqrt{3}$ .

← Back to all problems