HMMT February 2024 #0kyp

HMMT February 2024Hard

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.

TangentsRadical axis theoremReflectionTranslationCyclic quadrilateralsAngle chasingTrigonometry

Problem

Problem:

Suppose point $P$ is inside quadrilateral $A B C D$ such that

\begin{aligned} & \angle P A B = \angle P D A \\ & \angle P A D = \angle P D C, \\ & \angle P B A = \angle P C B, \text{ and } \\ & \angle P B C = \angle P C D \end{aligned}

If $P A = 4$ , $P B = 5$ , and $P C = 10$ , compute the perimeter of $A B C D$ .

← Back to all problems