Found problems: 8
2021 Kyiv City MO Round 1, 7.4
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]
1985 Spain Mathematical Olympiad, 8
A square matrix is sum-magic if the sum of all elements in each row, column and major diagonal is constant. Similarly, a square matrix is product-magic if the product of all elements in each row, column and major diagonal is constant.
Determine if there exist $3\times 3$ matrices of real numbers which are both sum-magic and product-magic.
1989 Irish Math Olympiad, 2
A 3x3 magic square, with magic number $m$, is a $3\times 3$ matrix such that the entries on each row, each column and each diagonal sum to $m$. Show that if the square has positive integer entries, then $m$ is divisible by $3$, and each entry of the square is at most $2n-1$, where $m=3n$. An example of a magic square with $m=6$ is
\[\left( \begin{array}{ccccc}
2 & 1 & 3\\
3 & 2 & 1\\
1 & 3 & 2
\end{array} \right)\]
2021 Kyiv City MO Round 1, 8.3
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]
2001 Estonia National Olympiad, 5
A $3\times 3$ table is filled with real numbers in such a way that each number in the table is equal to the absolute value of the difference of the sum of numbers in its row and the sum of numbers in its column.
(a) Show that any number in this table can be expressed as a sum or a difference of some two numbers in the table.
(b) Show that there is such a table not all of whose entries are $0$.
1996 Estonia National Olympiad, 3
Numbers $1992,1993, ... ,2000$ are written in a $3 \times 3$ table to form a magic square (i.e. the sums of numbers in rows, columns and big diagonals are all equal). Prove that the number in the center is $1996$. Which numbers are placed in the corners?
1992 Chile National Olympiad, 7
$\bullet$ 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.
1998 Tournament Of Towns, 3
Nine numbers are arranged in a square table:
$a_1 \,\,\, a_2 \,\,\,a_3$
$b_1 \,\,\,b_2 \,\,\,b_3$
$c_1\,\,\, c_2 \,\,\,c_3$ .
It is known that the six numbers obtained by summing the rows and columns of the table are equal:
$a_1 + a_2 + a_3 = b_1 + b_2 + b_3 = c_1 + c_2 + c_3 = a_1 + b_1 + c_1 = a_2 + b_2 + c_2 = a_3 + b_3 + c_3$ .
Prove that the sum of products of numbers in the rows is equal to the sum of products of numbers in the columns:
$a_1 b_1 c_1 + a_2 b_2c_2 + a_3b_3c_3 = a_1a_2a_3 + b_1 b_2 b_3 + c_1 c_2c_3$ .
(V Proizvolov)