Problems

Does there exist a function $f(x)$ defined for all $x\in \mathbb{R}$ and for all $x,y\in \mathbb{R}$ satisfying the inequality $|f(x+y)+\sin x+\sin y|<2$?

A square grid on the plane and a triangle with vertices at the nodes of the grid are given. Prove that the tangent of any angle in the triangle is a rational number.

Prove that the function $\cos \sqrt{x}$ is not periodic.

Solve the equation $2\sin \pi x/2-2\cos \pi x=x^5+10x-54$.

Prove that the sequence $x_n=\sin (n^2)$ does not tend to zero for $n$ that tends to infinity.

In the centre of a rectangular billiard table that is 3 m long and 1 m wide, there is a billiard ball. It is hit by a cue in a random direction. After the impact the ball stops passing exactly 2 m. Find the expected number of reflections from the sides of the table.

The number $x$ is such that both the sums $S=\sin 64x+\sin 65x$ and $C=\cos 64x+\cos 65x$ are rational numbers.

Prove that in both of these sums, both terms are rational.

Find the locus of points whose coordinates $(x,y)$ satisfy the relation $\sin (x+y)=0$.