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2017 Polish Junior Math Olympiad First Round P3

Source: 2017 Polish Junior Math Olympiad First Round

May 23, 2023
combinatorics

Problem Statement

In each square of an 11×1111\times 11 board, we are to write one of the numbers 1-1, 00, or 11 in such a way that the sum of the numbers in each column is nonnegative and the sum of the numbers in each row is nonpositive. What is the smallest number of zeros that can be written on the board? Justify your answer.