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f(x) = ax^2 + bx + c is divisible by prime p whenever x , 0 < a, b, c <= p

Source: 2023 NZMO - New Zealand Maths Olympiad Round 1 p4

September 2, 2023
number theory

Problem Statement

Let pp be a prime and let f(x)=ax2+bx+cf(x) = ax^2 + bx + c be a quadratic polynomial with integer coefficients such that 0<a,b,c≤p0 < a, b, c \le p. Suppose f(x)f(x) is divisible by pp whenever xx is a positive integer. Find all possible values of a+b+ca + b + c.
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