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1-100 in 10x10 grid

Source: VI Soros Olympiad 1990-00 R1 10.7 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

May 28, 2024
combinatoricscombinatorial geometry

Problem Statement

The numbers 1,2,3,...,99,1001, 2, 3, ..., 99, 100 are randomly arranged in the cells of a square table measuring 10×1010\times 10 (each number is used only once). Prove that there are three cells in the table whose sum of numbers does not exceed 18282. The centers of these cells form an isosceles right triangle, the legs of which are parallel to the edges of the table.
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