Harvard-MIT Mathematics Tournament #0j6t

Harvard-MIT Mathematics TournamentHard

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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.

φ (Euler's totient)Greatest common divisors (gcd)

Problem

Problem:

Let $n$ be a positive integer, and let $a_{1}, a_{2}, \ldots, a_{n}$ be a set of positive integers such that $a_{1}=2$ and $a_{m}=\varphi\left(a_{m+1}\right)$ for all $1 \leq m \leq n-1$ , where, for all positive integers $k$ , $\varphi(k)$ denotes the number of positive integers less than or equal to $k$ that are relatively prime to $k$ . Prove that $a_{n} \geq 2^{n-1}$ .

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