Harvard-MIT Mathematics Tournament #0j59

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.

Problem

Problem:

Denote $\{1,2, \ldots, n\}$ by $[n]$ , and let $S$ be the set of all permutations of $[n]$ . Call a subset $T$ of $S$ good if every permutation $\sigma$ in $S$ may be written as $t_{1} t_{2}$ for elements $t_{1}$ and $t_{2}$ of $T$ , where the product of two permutations is defined to be their composition. Call a subset $U$ of $S$ extremely good if every permutation $\sigma$ in $S$ may be written as $s^{-1} u s$ for elements $s$ of $S$ and $u$ of $U$ . Let $\tau$ be the smallest value of $|T| /|S|$ for all good subsets $T$ , and let $v$ be the smallest value of $|U| /|S|$ for all extremely good subsets $U$ . Prove that $\sqrt{v} \geq \tau$ .