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inequality with reals in (0,1)

Source: Problem 1, Polish NO 1988

October 16, 2005
inequalitiesfunctionalgebradomainanalytic geometryinequalities unsolved

Problem Statement

The real numbers x1,x2,...,xnx_1, x_2, ... , x_n belong to the interval (0,1)(0,1) and satisfy x1+x2+...+xn=m+rx_1 + x_2 + ... + x_n = m + r, where mm is an integer and r∈[0,1)r \in [0,1). Show that x12+x22+...+xn2≤m+r2x_1 ^2 + x_2 ^2 + ... + x_n ^2 \leq m + r^2.
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