Russian 97/1 solution

Problem 1

p(x) is a quadratic polynomial with non-negative coefficients. Show that p(xy)² ≤ p(x²)p(y²).

Solution

lhs = ( (√a x²)(√a y²) + (√b x)(√b y) + √c √c)² ≤ (ax⁴ + bx² + c)(ay⁴ + by² + c) = rhs by Cauchy-Schwartz.

Thanks to Suat Namli