Problems

Two ants crawled along their own closed route on a $7\times 7$ board. Each ant crawled only on the sides of the cells of the board and visited each of the 64 vertices of the cells exactly once. What is the smallest possible number of cell edges, along which both the first and second ants crawled?

101 random points are chosen inside a unit square, including on the edges of the square, so that no three points lie on the same straight line. Prove that there exist some triangles with vertices on these points, whose area does not exceed 0.01.

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 $f(2^{2011})$.

On a chessboard, $n$ white and $n$ black rooks are arranged so that the rooks of different colours cannot capture one another. Find the greatest possible value of $n$.

There are a thousand tickets with numbers 000, 001, ..., 999 and a hundred boxes with the numbers 00, 01, ..., 99. A ticket is allowed to be dropped into a box if the number of the box can be obtained from the ticket number by erasing one of the digits. Is it possible to arrange all of the tickets into 50 boxes?

2011 numbers are written on a blackboard. It turns out that the sum of any of these written numbers is also one of the written numbers. What is the minimum number of zeroes within this set of 2011 numbers?

Does there exist a real number $\alpha$ such that the number $\cos \alpha$ is irrational, and all the numbers $\cos 2\alpha$, $\cos 3\alpha$, $\cos 4\alpha$, $\cos 5\alpha$ are rational?

The sequence of numbers $a_1,a_2,\dots$ is given by the conditions $a_1=1$, $a_2=143$ and

for all $n\ge 2$.

Prove that all members of the sequence are integers.

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

We are given $n+1$ different natural numbers, which are less than $2n$ ($n>1$). Prove that among them there will always be three numbers, where the sum of two of them is equal to the third.