Problems

We call a number \(x\) rational if it can be represented as \(x=\frac{p}{q}\) for coprime integers \(p\) and \(q\). Otherwise we call the number irrational.
Non-zero numbers \(a\) and \(b\) satisfy the equality \(a^2b^2 (a^2b^2 + 4) = 2(a^6 + b^6)\). Prove that at least one of them is irrational.

Can the cells of a \(5 \times 5\) board be painted in 4 colours so that the cells located at the intersection of any two rows and any two columns are painted in at least three colours?

The real numbers \(x\) and \(y\) are such that for any distinct prime odd \(p\) and \(q\) the number \(x^p + y^q\) is rational. Prove that \(x\) and \(y\) are rational numbers.

Is it possible to arrange the numbers 1, 2, ..., 60 in a circle in such an order that the sum of every two numbers, between which lies one number, is divisible by 2, the sum of every two numbers between which lie two numbers, is divisible by 3, the sum of every two numbers between which lie six numbers, is divisible by 7?

The functions \(f (x) - x\) and \(f (x^2) - x^6\) are defined for all positive \(x\) and increase. Prove that the function

also increases for all positive \(x\).

Five teams participated in a football tournament. Each team had to play exactly one match with each of the other teams. Due to financial difficulties, the organisers cancelled some of the games. As a result, it turned out that all teams scored a different number of points and no team scored zero points. What is the smallest number of games that could be played in the tournament, if three points were awarded for a victory, one for a draw and zero for a defeat?

Looking back at her diary, Natasha noticed that in the date 17/02/2008 the sum of the first four numbers are equal to the sum of the last four. When will this coincidence happen for the last time in 2008?

A cinema contains 7 rows each with 10 seats. A group of 50 children went to see the morning screening of a film, and returned for the evening screening. Prove that there will be two children who sat in the same row for both the morning and the evening screening.

The numbers \(p\) and \(q\) are such that the parabolas \(y = - 2x^2\) and \(y = x^2 + px + q\) intersect at two points, bounding a certain figure.

Find the equation of the vertical line dividing the area of this figure in half.

100 queens, that cannot capture each other, are placed on a \(100 \times 100\) chessboard. Prove that at least one queen is in each \(50 \times 50\) corner square.