Berkeley Math Circle: Monthly Contest 6 #0hwd

Berkeley Math Circle: Monthly 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:

Suppose there are 100 prisoners, each of whom is given a number between 1 and 100. There is also a room with 100 boxes, labeled 1 to 100, and 100 pieces of paper with the numbers 1 to 100 on them. Each piece of paper is randomly placed inside one of the 100 boxes.

One at a time, each prisoner is allowed to enter the room and open at most 50 boxes to see the numbers inside. If every prisoner opens the box with his own number inside it, they will all be released. They are not allowed to communicate at all during this process, but they can come up with a strategy beforehand. Show that there is a strategy that gives them at least a $30\%$ chance of winning.