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Problem 846

AIME late

Combinatorics Difficulty 5.6 Prove it

$37 *$ . In each cell of an infinite sheet of graph paper, a real number is written such that the sum of the numbers inside any square with sides along the grid lines, in absolute value, does not exceed 1. Prove that the sum of the numbers inside any rectangle with sides along the grid lines does not exceed 10000.

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

70.37. The operation used in solving problem 70.38 (see below) applied to a rectangle of size $A \times B(A>B)$ allows obtaining a rectangle of size $|A-2 B| \times B$ . Prove that sooner or later we will get either a square or an "empty" rectangle.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.