MathDB
Rounding distances

Source: 14th XMO P1

January 14, 2024
inequalities

Problem Statement

Nonnegative reals x1x_1, x2x_2, …\dots, xnx_n satisfies x1+x2+⋯+xn=nx_1+x_2+\dots+x_n=n. Let ∣∣x∣∣||x|| be the distance from xx to the nearest integer of xx (e.g. ∣∣3.8∣∣=0.2||3.8||=0.2, ∣∣4.3∣∣=0.3||4.3||=0.3). Let yi=xi∣∣xi∣∣y_i = x_i ||x_i||. Find the maximum value of ∑i=1nyi2\sum_{i=1}^n y_i^2.
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