Problem

Source: Iran 2024 3rd Round Combinatorics Test P2

Tags: combinatorics, table, Tabular Triangle, cells



Consider the main diagonal and the cells above it in an \( n \times n \) grid. These cells form what we call a tabular triangle of length \( n \). We want to place a real number in each cell of a tabular triangle of length \( n \) such that for each cell, the sum of the numbers in the cells in the same row and the same column (including itself) is zero. For example, the sum of the cells marked with a circle is zero. It is known that the number in the topmost and leftmost cell is $1.$ Find all possible ways to fill the remaining cells.


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