HMMT February 2015 #0jr5

HMMT February 2015Hard

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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: Find the number of ordered pairs of integers $(a, b) \in \{1,2, \ldots, 35\}^2$ (not necessarily distinct) such that $a x+b$ is a "quadratic residue modulo $x^2+1$ and 35", i.e. there exists a polynomial $f(x)$ with integer coefficients such that either of the following equivalent conditions holds:

there exist polynomials $P, Q$ with integer coefficients such that $f(x)^2-(a x+b)=\left(x^2+1\right) P(x)+35 Q(x)$ ;
or more conceptually, the remainder when (the polynomial) $f(x)^2-(a x+b)$ is divided by (the polynomial) $x^2+1$ is a polynomial with (integer) coefficients all divisible by 35.