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Polynomial bounded by sum of coefficients (BxMO 2022, Problem 1)

Source: BxMO 2022, Problem 1

May 1, 2022
BxMOalgebrapolynomial

Problem Statement

Let n⩾0n\geqslant 0 be an integer, and let a0,a1,…,ana_0,a_1,\dots,a_n be real numbers. Show that there exists k∈{0,1,…,n}k\in\{0,1,\dots,n\} such that a0+a1x+a2x2+⋯+anxn⩽a0+a1+⋯+aka_0+a_1x+a_2x^2+\cdots+a_nx^n\leqslant a_0+a_1+\cdots+a_k for all real numbers x∈[0,1]x\in[0,1].
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