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2017 Polish Junior Math Olympiad Second Round P4

Source: 2017 Polish Junior Math Olympiad Second Round

May 22, 2023
number theory

Problem Statement

Do numbers x1,x2,…,x99x_1, x_2, \ldots, x_{99} exist, where each of them is equal to 2+1\sqrt{2}+1 or 2−1\sqrt{2}-1, and satisfy the equation x1x2+x2x3+x3x4+…+x98x99+x99x1=199 ?x_1x_2+x_2x_3+x_3x_4+\ldots+x_{98}x_{99}+x_{99}x_1=199\,? Justify your answer.
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