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Find pair of indices for which inequality holds

Source: Ukrainian Mathematical Olympiad 2021. Day 1, Problem 9.3

December 21, 2023

inequalities

Problem Statement

Nonnegative integers

x_1, x_2 , \ldots, x_n

are such that

x_1^2 + x_2^2 + \ldots +x_n^2 = n

n > 1

. Prove that for any nonnegative real numbers

y_1, y_2 , \ldots, y_n

there exist indices

i, j \in \{1, 2, \ldots, n\}

, not necessarily different, for which the following inequality holds:

x_i + y_j - x_{j+1}y_{i+1} \geq 1

(Here index

n+1 =

index

1

).
Proposed by Nazar Serdyuk