Found problems: 59
Each of the $16$ squares in a $4 \times 4$ table contains a number. For any square, the sum of the numbers in the squares sharing a common side with the chosen square is equal to $1$. Determine the sum of all $16$ numbers in the table.
(R Zhenodarov)
Can the squares of a $1990 \times 1990$ chessboard be colored black or white so that half the squares in each row and column are black and cells symmetric with respect to the center are of opposite color?
The numbers $1, 2, 3, ... , 64$ are written in the cells of an $8 \times 8$ board (in some order, one per cell). Show that at least four $2 \times 2$ squares have sum greater than $100$.
Given a table $4\times 4$.
a) Find, how $7$ stars can be put in its fields in such a way, that erasing of two arbitrary lines and two columns will always leave at list one of the stars.
b) Prove that if there are less than $7$ stars, You can always find two columns and two rows, such, that if you erase them, no star will remain in the table.
In a grid of dimensions $n \times n$, a part of the squares is marked with crosses such that in each at least half of the $4 \times 4$ squares are marked. Find the least possible the total number of marked squares in the grid.
Twenty-five people of different heights stand in a $5\times 5$ grid of squares, with one person in each square. We know that each row has a shortest person, suppose Ana is the tallest of these five people. Similarly, we know that each column has a tallest person, suppose Bev is the shortest of these five people.
Assuming Ana and Bev are not the same person, who is taller: Ana or Bev?
Prove that your answer is always correct.
$n^2$ real numbers are written in a square $n \times n$ table so that the sum of the numbers in each row and column equals zero. A move is to add a row to one column and subtract it from another (so if the entries are $a_{ij}$ and we select row $i$, column $h$ and column $k$, then column h becomes $a_{1h} + a_{i1}, a_{2h} + a_{i2}, ... , a_{nh} + a_{in}$, column $k$ becomes $a_{1k} - a_{i1}, a_{2k} - a_{i2}, ... , a_{nk} - a_{in}$, and the other entries are unchanged). Show that we can make all the entries zero by a series of moves.
All the fields of a square $n\times n$ (n>2) table are filled with $+1$ or $-1$ according to the rules:
[i]At the beginning $-1$ are put in all the boundary fields. The number put in the field in turn (the field is chosen arbitrarily) equals to the product of the closest, from the different sides, numbers in its row or in its column.
[/i]
a) What is the minimal
b) What is the maximal
possible number of $+1$ in the obtained table?
Peter marks several cells on a $5\times 5$ board. Basil wins if he can cover all marked cells with three-cell corners. The corners must be inside the board and not overlap. What is the least number of cells Peter should mark to prevent Basil from winning? (Cells of the corners must coincide with the cells of the board).