Is it possible to fill an \(n\times n\) table with the numbers \(-1\), \(0\), \(1\), such that the sums of all the rows, columns, and diagonals are unique?
Prove that in any group of friends there will be two people who have the same number of friends.
In chess, ‘check’ is when the king is under threat of capture from another piece. What is the largest number of kings that it is possible to place on a standard \(8\times 8\) chess board so that no two check one another.
Solve the equation \(2x^x = \sqrt {2}\) for positive numbers.
Find the first 99 decimal places in the number expansion of \((\sqrt{26} + 5)^{99}\).
Let \(M\) be a finite set of numbers. It is known that among any three of its elements there are two, the sum of which belongs to \(M\).
What is the largest number of elements in \(M\)?
Each of the three cutlets should be fried in a pan on both sides for five minutes each side. Only two cutlets can fit onto the frying pan. Is it possible to fry all three cutlets more quickly than in 20 minutes (if the time to turn over and transfer the cutlets is neglected)?
An area of airspace contains clouds. It turns out that the area can be divided by 10 aeroplanes into regions such that each region contains no more than one cloud. What is the largest number of clouds an aircraft can fly through whilst holding a straight line course.
A standard chessboard has more than a quarter of its squares filled with chess pieces. Prove that at least two adjacent squares, either horizontally, vertically, or diagonally, are occupied somewhere on the board.
A convex figure and point \(A\) inside it are given. Prove that there is a chord (that is, a segment joining two boundary points of a convex figure) passing through point \(A\) and dividing it in half at point \(A\).