Problems

Find the sum $1/3+2/3+2^2/3+2^3/3+\cdots +2^{1000}/3$.

Suppose that there are 15 prime numbers forming an arithmetic progression with a difference of $d$. Prove that $d>30,000$.

There are one hundred natural numbers, they are all different, and sum up to 5050. Can you find those numbers? Are they unique, or is there another bunch of such numbers?

In a volleyball tournament teams play each other once. A win gives the team 1 point, a loss 0 points. It is known that at one point in the tournament all of the teams had different numbers of points. How many points did the team in second last place have at the end of the tournament, and what was the result of its match against the eventually winning team?

$A$ and $B$ shoot in a shooting gallery, but they only have one six-shot revolver with one cartridge. Therefore, they agreed in turn to randomly rotate the drum and shoot. $A$ goes first. Find the probability that the shot will occur when $A$ has the revolver.

At the Antarctic station, there are $n$ polar explorers, all of different ages. With the probability $p$ between each two polar explorers, friendly relations are established, regardless of other sympathies or antipathies. When the winter season ends and it’s time to go home, in each pair of friends the senior gives the younger friend some advice. Find the mathematical expectation of the number of those who did not receive any advice.

At the ball, there were $n$ married couples. In each pair, the husband and wife are of the same height, but there are no two pairs of the same height. The waltz begins, and all those who came to the ball randomly divide into pairs: each gentleman dances with a randomly chosen lady.

Find the mathematical expectation of the random variable $X$, “the number of gentlemen who are shorter than their partners”.

Is the sum of the numbers $1+2+3+\cdots +1999$ divisible by 1999?

The function $f(x)$ is defined on the positive real $x$ and takes only positive values. It is known that $f(1)+f(2)=10$ and $f(a+b)=f(a)+f(b)+2\sqrt{f(a)f(b)}$ for any $a$ and $b$. Find $f(2^{2011})$.

How many integers are there from 0 to 999999, in the decimal notation of which there are no two identical numbers next to each other?