IMO 1969 LL GBR24

Origin: GBR

Problem

The polynomial P(x) = a0xk + a1xk−1 + \cdot \cdot \cdot + ak, where a0, . . . , ak are integers, is said to be divisible by an integer m if P(x) is a multiple of m for every integral value of x. Show that if P(x) is divisible by m, then a0 \cdotk! is a multiple of m. Also prove that if a, k, m are positive integers such that ak! is a multiple of m, then a polynomial P(x) with leading term axk can be found that is divisible by m.