IMO 1986 LL GBR34

For each nonnegative integer n, Fn(x) is a polynomial in x of

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IMO 1986 LL GBR34

Origin: GBR

Problem

For each nonnegative integer n, Fn(x) is a polynomial in x of degreee n. Prove that if the identity Fn(2x) = n  r=0 (−1)n−r n r  2rFr(x) holds for each n, then Fn(tx) = n  r=0 n r  tr(1 −t)n−rFr(x) for each n and all t.