Berkeley Math Circle Take-Home Contest #6 #0huf

Berkeley Math Circle Take-Home Contest #6Hard

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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:

Acute triangle $A B C$ is made of solid metal, and it is on top of a wooden table. Points $P$ on $A B$ and $Q$ on $A C$ are such that the perpendicular to $A B$ through $P$ intersects the perpendicular to $A C$ through $Q$ inside the triangle. Nails are hammered into the table at $P$ and $Q$ . (The nails do not go through the triangle but are at its edges, bounding the triangle.) Prove that there is a unique position $R$ on $B C$ such that, if a nail is hammered into the table at $R$ , the triangle will no longer be able to move (within the plane of the table).