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f(x_1, y_1) = f(x_2, y_2) = ....= f(x_{2023}, y_{2023}) , f(x, y) <= xy

Source: 2023 Swedish Mathematical Competition p4

March 24, 2024
algebracombinatorics

Problem Statement

Let ff be a function that associates a positive integer (x,y)(x, y) with each pair of positive integers f(x,y)f(x, y). Suppose that f(x,y)≤xyf(x, y) \le xy for all positive integers xx, yy. Show that there are 20232023 different pairs (x1,y1)(x_1, y_1),......, (x2023,y2023 (x_{2023}, y_{2023}) such that f(x1,y1)=f(x2,y2)=....=f(x2023,y2023).f(x_1, y_1) = f(x_2, y_2) = ....= f(x_{2023}, y_{2023}).
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