Problems

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?

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

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 $f(2^{2011})$.

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

$A$ 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 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?