IMO 1976 SL 8

Origin: SWE

Problem

Let P be a polynomial with real coefficients such that P(x) > 0 if x > 0. Prove that there exist polynomials Q and R with nonnegative coefficients such that P(x) = Q(x) R(x) if x > 0.

Solution

Every polynomial with real coefficients can be factored as a product of linear and quadratic polynomials with real coefficients. Thus it suffices to prove the result only for a quadratic polynomial P(x) = x2 −2ax + b2, with a > 0 and b2 > a2. Using the identity (x2 + b2)2n −(2ax)2n = (x2 −2ax + b2) 2n−1  k=0 (x2 + b2)k(2ax)2n−k−1 we have solved the problem if we can choose n such that b2n
2n n 

22na2n. However, it is is easy to show that 2n
2n n  < 22n; hence it is enough to take n such that (b/a)2n > 2n. Since limn\to\infty(2n)1/(2n) = 1 < b/a, such an n always exists.