There are real numbers written on each field of a \(m \times n\) chessboard. Some of them are negative, some are positive. In one move we can multiply all the numbers in one column or row by \(-1\). Is that always possible to obtain a chessboard where sums of numbers in each row and column are nonnegative?
A \(7 \times 7\) square was tiled using \(1 \times 3\) rectangular blocks. One of the squares has not been covered. Which one can it be?
Tom found a large, old clock face and put 12 sweets on the number 12. Then he started to play a game with himself. In each move he moves one sweet to the next number clockwise, and some other to the next number anticlockwise. Is it possible that after finite number of steps there is exactly 1 of the sweets on each number?
Is it possible for the mean of some 35 whole numbers to equal \(6.35\)?
Does there exist a natural number which, when divided by the sum of its digits, gives a quotient and remainder both equal to the number 2011?
Is it true that, if \(b>a+c>0\), then the quadratic equation \(ax^2 +bx+c=0\) has two roots?
The digits of a 3 digit number \(A\) were written in reverse order and this is the number \(B\). Is it possible to find a value of \(A\) such that the sum of \(A\) and \(B\) has only odd numbers as its digits?
Let \(x\) be a 2 digit number. Let \(A\), \(B\) be the first (tens) and second (units) digits of \(x\), respectively. Suppose \(A\) is twice as large as \(B\). If we add the square of \(A\) to \(x\) then we get the square of a certain whole number. Find the value of \(x\).
A ream of squared paper is shaded in two colours. Prove that there are two horizontal and two vertical lines, the points of intersection of which are shaded in the same colour.
There are 25 points on a plane, and among any three of them there can be found two points with a distance between them of less than 1. Prove that there is a circle of radius 1 containing at least 13 of these points.