Ukrainian National Mathematical Olympiad

Question

Numbers a, b, c, d satisfy: ab + cd > 0, ac + bd > 0, and a^2 + d^2 = c^2 + b^2. Prove that ad + bc > 0.

Accepted Answer

(ab + cd)(ac + bd) = a^2bc + ac^2d + ab^2d + bcd^2 = bc(a^2 + d^2) + ad(c^2 + b^2) = bc(a^2 + d^2) + ad(a^2 + d^2) = (a^2 + d^2)(bc + ad) > 0, which implies the required inequality.