Berkeley Math Circle Monthly Contest 6 #0hrz

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:

The Moria Indestructible Phone Co. has hired you to test the hardiness of their newest smartphone model, the Mithril II. Your assignment is to determine the lowest floor of the Burj Khalifa tower (the world's tallest building, with 163 floors) from which the phone must be dropped to break it. You can ride the elevator to any floor, drop the phone to the ground, and then test whether it is intact. You may assume that if the phone breaks at a given floor, it consistently breaks at that floor and all higher floors. But the company has given you only two Mithril II's to test, and once one of them breaks, it remains broken. What is the minimum number of drops needed to determine the minimum floor of breaking, or else to conclude that the phone will withstand dropping from any of the floors?