Problems

A castle is surrounded by a circular wall with nine towers, at which there are knights on duty. At the end of each hour, they all move to the neighbouring towers, each knight moving either clockwise or counter-clockwise. During the night, each knight stands for some time at each tower. It is known that there was an hour when at least two knights were on duty at each tower, and there was an hour when there was precisely one knight on duty on each of exactly five towers. Prove that there was an hour when there were no knights on duty on one of the towers.

In the entry $\ast +\ast +\ast +\ast +\ast +\ast +\ast +\ast =\ast \ast$ replace the asterisks with different digits so that the equality is correct.

Hannah placed 101 counters in a row which had values of 1, 2 and 3 points. It turned out that there was at least one counter between every two one point counters, at least two counters lie between every two two point counters, and at least three counters lie between every two three point counters. How many three point counters could Hannah have?

A chequered strip of $1\times N$ is given. Two players play the game. The first player puts a cross into one of the free cells on his turn, and subsequently the second player puts a nought in another one of the cells. It is not allowed for there to be two crosses or two noughts in two neighbouring cells. The player who is unable to make a move loses.

Which of the players can always win (no matter how their opponent played)?

We are given 111 different natural numbers that do not exceed 500. Could it be that for each of these numbers, its last digit coincides with the last digit of the sum of all of the remaining numbers?

The number $x$ is such a number that exactly one of the four numbers $a=x-\sqrt{2}$, $b=x-1/x$, $c=x+1/x$, $d=x^2+2\sqrt{2}$ is not an integer. Find all such $x$.

The numbers $x$, $y$ and $z$ are such that all three numbers $x+yz$, $y+zx$ and $z+xy$ are rational, and $x^2+y^2=1$. Prove that the number $xy{z}^2$ is also rational.

Peter marks several cells on a $5\times 5$ board. His friend, Richard, will win if he can cover all of these cells with non-overlapping corners of three squares, that do not overlap with the border of the square (you can only place the corners on the squares). What is the smallest number of cells that Peter should mark so that Richard cannot win?

16 teams took part in a handball tournament where a victory was worth 2 points, a draw – 1 point and a defeat – 0 points. All teams scored a different number of points, and the team that ranked seventh, scored 21 points. Prove that the winning team drew at least once.

In the isosceles triangle $ABC$, the angle $B$ is equal to ${30}^{\circ }$, and $AB=BC=6$. The height $CD$ of the triangle $ABC$ and the height $DE$ of the triangle $BDC$ are drawn. Find the length $BE$.