HMMT February 2024 #0kyx

HMMT February 2024Hard

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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.

PolynomialsVieta's formulasPolynomials mod pChinese remainder theoremQuadratic residuesQuadratic reciprocity

Problem

Problem:

A polynomial $f \in \mathbb{Z}[x]$ is called splitty if and only if for every prime $p$ , there exist polynomials $g_p, h_p \in \mathbb{Z}[x]$ with $\operatorname{deg} g_p, \operatorname{deg} h_p < \operatorname{deg} f$ and all coefficients of $f - g_p h_p$ are divisible by $p$ . Compute the sum of all positive integers $n \leq 100$ such that the polynomial $x^4 + 16 x^2 + n$ is splitty.

Proposed by: Pitchayut Saengrungkongka

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