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Every polynomial P(x) of degree 99

Source: Tuymaada 2009, Senior League, Second Day, Problem 4

July 19, 2009
algebrapolynomialfloor functionfunctional equationalgebra unsolved

Problem Statement

Determine the maximum number h h satisfying the following condition: for every a∈[0,h] a\in [0,h] and every polynomial P(x) P(x) of degree 99 such that P(0)\equal{}P(1)\equal{}0, there exist x1,x2∈[0,1] x_1,x_2\in [0,1] such that P(x_1)\equal{}P(x_2) and x_2\minus{}x_1\equal{}a. Proposed by F. Petrov, D. Rostovsky, A. Khrabrov
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