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IMO LongList 1987 Polynomials

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September 6, 2010

algebrapolynomialalgebra unsolved

Problem Statement

Let

P,Q,R

be polynomials with real coefficients, satisfying

P^4+Q^4 = R^2

. Prove that there exist real numbers

p, q, r

and a polynomial

S

such that

P = pS, Q = qS

and

R = rS^2

.
[hide="Variants"]Variants. (1)

P^4 + Q^4 = R^4

; (2)

\gcd(P,Q) = 1

; (3)

\pm P^4 + Q^4 = R^2

or

R^4.

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