This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 3

2021 Kyiv City MO Round 1, 8.3
Tags: permutations , Magic squares , combinatorics
The $1 \times 1$ cells located around the perimeter of a $3 \times 3$ square are filled with the numbers $1, 2, \ldots, 8$ so that the sums along each of the four sides are equal. In the upper left corner cell is the number $8$, and in the upper left is the number $6$ (see the figure below). [img]https://i.ibb.co/bRmd12j/Kyiv-MO-2021-Round-1-8-2.png[/img] How many different ways to fill the remaining cells are there under these conditions? [i]Proposed by Mariia Rozhkova[/i]

1992 Chile National Olympiad, 7
Tags: combinatorics , Magic squares , Sum , board
$\bullet$ Determine a natural $n$ such that the constant sum $S$ of a magic square of $ n \times n$ (that is, the sum of its elements in any column, or the diagonal) differs as little as possible from $1992$. $\bullet$ Construct or describe the construction of this magic square.

2021 Kyiv City MO Round 1, 7.4
Tags: permutations , rectangle , Magic squares
A rectangle $3 \times 5$ is divided into $15$ $1 \times 1$ cells. The middle $3$ cells that have no common points with the border of the rectangle are deleted. Is it possible to put in the remaining $12$ cells numbers $1, 2, \ldots, 12$ in some order, so that the sums of the numbers in the cells along each of the four sides of the rectangle are equal? [i]Proposed by Mariia Rozhkova[/i]