IMC 2008 · Day 2 · P10

Let $f(x) = g(x) h(x)$ where $h(x)$ is a polynomial with integer coefficients.

Let $a_1, \dots, a_{81}$ be distinct integer roots of the polynomial $f(x) - 2008$. Then $f(a_i) = g(a_i) h(a_i) = 2008$ for $i = 1, \dots, 81$, Hence, $g(a_1), \dots, g(a_{81})$ are integer divisors of 2008.

Since $2008 = 2^3 \cdot 251$ (2, 251 are primes) then 2008 has exactly 16 distinct integer divisors (including the negative divisors as well). By the pigeonhole principle, there are at least 6 equal numbers among $g(a_1), \dots, g(a_{81})$ (because $81 > 16 \cdot 5$). For example, $g(a_1) = g(a_2) = \dots = g(a_6) = c$. So $g(x) - c$ is a nonconstant polynomial which has at least 6 distinct roots (namely $a_1, \dots, a_6$). Then the degree of the polynomial $g(x) - c$ is at least 6.