Berkeley Math Circle Monthly Contest #1 #0i0z

Berkeley Math Circle Monthly Contest #1Hard

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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:

There are 30 members of the math faculty at UC Berkeley. The evil Gastropod places a red or green stamp on each member's forehead, then places them all in separate rooms. Each day, the Gastropod convenes all the faculty in one room and asks any member who has conclusively determined that his/her stamp is red to raise his/her hand. No other communication between them is allowed. As it turns out, everyone has a red stamp (but since each can only see the others' stamps, none of them realize this at first). The Gastropod remarks on the first day: "At least one of you has a red stamp." Given that all Berkeley faculty members are perfect reasoners, and each knows all the others are perfect reasoners, prove that everyone will eventually determine that his/her stamp is red.