IMO 1987 LL POL50

Let P, Q, R be polynomials with real coefficients, satisfying

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IMO 1987 LL POL50

Origin: POL

Problem

Let P, Q, R be polynomials with real coefficients, satisfying P 4+Q4 = R2. Prove that there exist real numbers p, q, r and a polynomial S such that P = pS, Q = qS and R = rS2. Variants: (1) P 4 + Q4 = R4; (2) gcd(P, Q) = 1; (3) \pmP 4 + Q4 = R2 or R4.