At a round table, 10 boys and 15 girls were seated. It turned out that there are exactly 5 pairs of boys sitting next to each other.
How many pairs of girls are sitting next to each other?
Solve the equation \(2x^x = \sqrt {2}\) for positive numbers.
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 \(M\)?
Two players play the following game. They take turns. One names two numbers that are at the ends of a line segment. The next then names two other numbers, which are at the ends of a segment nested in the previous one. The game goes on indefinitely. The first aims to have at least one rational number within the intersection of all of these segments, and the second aims to prevent such occurring. Who wins in this game?
Prove that rational numbers from \([0; 1]\) can be covered by a system of intervals of total length no greater than \(1/1000\).
The positive irrational numbers \(a\) and \(b\) are such that \(1/a + 1/b = 1\). Prove that among the numbers \(\lfloor ma\rfloor , \lfloor nb\rfloor\) each natural number occurs exactly once.
A rectangular billiard with sides 1 and \(\sqrt {2}\) is given. From its angle at an angle of \(45 ^\circ\) to the side a ball is released. Will it ever get into one of the pockets? (The pockets are in the corners of the billiard table).
Suppose that \(n \geq 3\). Are there n points that do not lie on one line, whose pairwise distances are irrational, and the areas of all of the triangles with vertices in them are rational?
Do there exist three points \(A\), \(B\) and \(C\) on the plane such that for any point \(X\) the length of at least one of the segments \(XA\), \(XB\) and \(XC\) is irrational?
Ten circles are marked on the circle. How many non-closed non-self-intersecting nine-point broken lines exist with vertices at these points?