HMIC #0ke0

HMICHard

From module Generating Functions (Intro)

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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.

Polynomials mod pFermat / Euler / Wilson theoremsQuadratic residuesGenerating functionsAlgebraic properties of binomial coefficients

Problem

Problem: Let $C_{k} = \frac{1}{k+1} \binom{2k}{k}$ denote the $k^{\text{th}}$ Catalan number and $p$ be an odd prime. Prove that exactly half of the numbers in the set

\left\{\sum_{k=1}^{p-1} C_{k} n^{k} \mid n \in \{1,2, \ldots, p-1\}\right\}

are divisible by $p$ .

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