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.

There are 30 students in the class. Prove that the probability that some two students have the same birthday is more than 50%.

Is it possible to:

a) load two coins so that the probability of “heads” and “tails” were different, and the probability of getting any of the combinations “tails, tails,” “heads, tails”, “heads, heads” be the same?

b) load two dice so that the probability of getting any amount from 2 to 12 would be the same?

The point $O$, lying inside the triangle $ABC$, is connected by segments with the vertices of the triangle. Prove that the variance of the set of angles $AOB$, $AOC$ and $BOC$ is less than a) $10{\pi }^2/27$; b) $2{\pi }^2/9$.

A high rectangle of width 2 is open from above, and the L-shaped domino falls inside it in a random way (see the figure).

a) $k$ $L$-shaped dominoes have fallen. Find the mathematical expectation of the height of the resulting polygon.

b) $7$ $G$-shaped dominoes fell inside the rectangle. Find the probability that the resulting figure will have a height of 12.