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P(x, y) = (x^2 + y^2)^k when P(sin t, cos t) = 1

Source: Estonia IMO TST 2010 p5

April 1, 2020

polynomialalgebratrigonometry

Problem Statement

Let

P(x, y)

be a non-constant homogeneous polynomial with real coefficients such that

P(\sin t, \cos t) = 1

for every real number

t

. Prove that there exists a positive integer

k

such that

P(x, y) = (x^2 + y^2)^k