Problems

Let $f(x)$ be a polynomial about which it is known that the equation $f(x)=x$ has no roots. Prove that then the equation $f(f(x))=x$ does not have any roots.

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 $M$?

Let $f$ be a continuous function defined on the interval $[0;1]$ such that $f(0)=f(1)=0$. Prove that on the segment $[0;1]$ there are 2 points at a distance of 0.1 at which the function $f4(x)$ takes equal values.

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 $A$.

Calculate ${\int }_0^{\pi /2}({\sin }^2(\sin x)+{\cos }^2(\cos x))dx$.

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.