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Number of cool polynomials is even

Source: 2024 Israel TST Test 1 P3

August 29, 2023
algebrapolynomialnumber theorymodular arithmeticprime numbers

Problem Statement

Let nn be a positive integer and pp be a prime number of the form 8k+58k+5. A polynomial QQ of degree at most 20232023 and nonnegative integer coefficients less than or equal to nn will be called "cool" if p∣Q(2)⋅Q(3)⋅…⋅Q(p−2)−1.p\mid Q(2)\cdot Q(3) \cdot \ldots \cdot Q(p-2)-1. Prove that the number of cool polynomials is even.
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