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 $|f(x+y)+\sin x+\sin y|<2$?

Prove the irrationality of the following numbers:

a) $\sqrt{3}17$

b) $\sqrt{2}+\sqrt{3}$

c) $\sqrt{2}+\sqrt{3}+\sqrt{5}$

d) $\sqrt{3}3-\sqrt{2}$

e) $\cos {10}^{\circ }$

f) $\tan {10}^{\circ }$

g) $\sin 1^{\circ }$

h) ${\log }_{2}3$

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.

Find the largest and smallest values of the functions

a) $f_1(x)=a\cos x+b\sin x$; b) $f_2(x)=a{\cos }^{2}x+b\cos x\sin x+c{\sin }^{2}x$.

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

Solve the system of equations: $$\begin{array}{rlrl}\sin y-\sin x& =x-y;& & \text{and}\\ \sin y-\sin z& =z-y;& & \text{and}\\ x-y+z& =\pi .\end{array}$$

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.

The numbers $a$ and $b$ are such that the first equation of the system $$\begin{array}{rl}\sin x+a& =bx\\ \cos x& =b\end{array}$$ has exactly two solutions. Prove that the system has at least one solution.

The numbers $a$ and $b$ are such that the first equation of the system $$\begin{array}{rl}\cos x& =ax+b\\ \sin x+a& =0\end{array}$$ has exactly two solutions. Prove that the system has at least one solution.