HMMT February 2016 #0jvv

HMMT February 2016Hard

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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.

DeterminantsMatricesRoots of unity

Problem

Problem: For any positive integer $n$ , $S_{n}$ be the set of all permutations of $\{1,2,3, \ldots, n\}$ . For each permutation $\pi \in S_{n}$ , let $f(\pi)$ be the number of ordered pairs $(j, k)$ for which $\pi(j)>\pi(k)$ and $1 \leq j<k \leq n$ . Further define $g(\pi)$ to be the number of positive integers $k \leq n$ such that $\pi(k) \equiv k \pm 1(\bmod n)$ . Compute

\sum_{\pi \in S_{999}}(-1)^{f(\pi)+g(\pi)}

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