Problems

At all rational points of the real line, integers are arranged. Prove that there is a segment such that the sum of the numbers at its ends does not exceed twice the number on its middle.

Prove that for any positive integer $n$ the inequality

is true.

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 $x$.

Prove that if the numbers $x,y,z$ satisfy the following system of equations for some values of $p$ and $q$: $$\begin{array}{rl}y& =x^2+px+q,\\ z& =y^2+py+q,\\ x& =z^2+pz+q,\end{array}$$ then the inequality $x^{2}y+y^{2}z+z^{2}x\ge x^{2}z+y^{2}x+z^{2}y$ is satisfied.

Find the largest natural number $n$ which satisfies $n^{200}<5^{300}$.

Of the four inequalities $2x>70$, $x<100$, $4x>25$ and $x>5$, two are true and two are false. Find the value of $x$ if it is known that it is an integer.

Solve the inequality: $\lfloor x\rfloor \times \{x\}<x-1$.

Prove that $\frac{1}{2}(x^2+y^2)\ge xy$ for any $x$ and $y$.

Prove that for $x\ge 0$ the inequality is valid: $2x+\frac{3}{8}\ge \sqrt[4]{x}$.

On a plane, there are 1983 points and a circle of unit radius. Prove that there is a point on the circle, from which the sum of the distances to these points is no less than 1983.