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Roots of polynomial composition

Source: Spain MO 2023 P4

March 12, 2023
algebrapolynomialSpain

Problem Statement

Let x1≤x2≤x3≤x4x_1\leq x_2\leq x_3\leq x_4 be real numbers. Prove that there exist polynomials of degree two P(x)P(x) and Q(x)Q(x) with real coefficients such that x1x_1, x2x_2, x3x_3 and x4x_4 are the roots of P(Q(x))P(Q(x)) if and only if x1+x4=x2+x3x_1+x_4=x_2+x_3.
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