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ASU 065 All Russian MO 1965 quasi-rounding n numbers

Source:

June 19, 2019
combinatoricsalgebra

Problem Statement

Quasi-rounding is a substitution one of the two closest integers instead of the given number. Given nn numbers. Prove that you can quasi-round them in such a way, that a sum of every subset of quasi-rounded numbers will deviate from the sum of the same subset of initial numbers not greater than (n+1)/4(n+1)/4 .