Problem

Source: 2024 Israel National Olympiad (Gillis) P3

Tags: combinatorics, Array, matchings, national olympiad



A triangle is composed of circular cells arranged in $5784$ rows: the first row has one cell, the second has two cells, and so on (see the picture). The cells are divided into pairs of adjacent cells (circles touching each other), so that each cell belongs to exactly one pair. A pair of adjacent cells is called diagonal if the two cells in it aren't in the same row. What is the minimum possible amount of diagonal pairs in the division? An example division into pairs is depicted in the image.


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