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u_1(x)^n +u_2(x)^n = u_3(x)^n, ui(x) = a_ix+b_i

Source: Polish MO Finals 1972 p1

August 22, 2024
polynomialalgebra

Problem Statement

Polynomials ui(x)=aix+biu_i(x) = a_ix+b_i (ai,bi∈Ra_i,b_i \in R, i=1,2,3 i = 1,2,3) satisfy u1(x)n+u2(x)n=u3(x)nu_1(x)^n +u_2(x)^n = u_3(x)^n for some integer n≥2.n \ge 2. Prove that there exist real numbers AA,BB,c1c_1,c2c_2,c3c_3 such that ui(x)=ci(Ax+B)u_i(x) = c_i(Ax+B) for i=1,2,3i = 1,2,3.
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