Problems

For how many pairs of numbers $x$ and $y$ between $1$ and $100$ is the expression $x^2+y^2$ divisible by $7$?

Seven robbers are dividing a bag of coins of various denominations. It turned out that the sum could not be divided equally between them, but if any coin is set aside, the rest could be divided so that every robber would get an equal part. Prove that the bag cannot contain $100$ coins.

Show that the equation $x^2+6x-1=y^2$ has no solutions in integer $x$ and $y$.

Draw how Robinson Crusoe should put pegs and ropes to tie his goat in order for the goat to graze grass in the shape of a square, or slightly harder in a shape of a given rectangle.

Draw how Robinson Crusoe should put pegs and ropes to tie his goat in order for the goat to graze grass in the shape of a given triangle.

Draw how Robinson Crusoe should put pegs and ropes to tie his goat in order for the goat to graze grass in the shape of a shape like this

Draw a picture how Robinson used to tie the goat and the wolf in order for the goat to graze the grass in the shape of half a circle.

A whole number of litres of water were poured into three vessels. You can only to pour into any vessel the exact amount of water equal to the amount it already contains from any other vessel. Prove that in a few transfusions one can empty one of the vessels. The vessels are large enough: each can hold all the water.

On a $10\times 10$ board, a bacterium sits in one of the cells. In one move, the bacterium shifts to a cell adjacent to the side (i.e. not diagonal) and divides into two bacteria (both remain in the same new cell). Then, again, one of the bacteria sitting on the board shifts to a new adjacent cell, either horizontally or vertically, and divides into two, and so on. Is it possible for there to be an equal number of bacteria in all cells after several such moves?

Let $a$, $b$ and $c$ be the three side lengths of a triangle. Does there exist a triangle with side lengths $a+1$, $b+1$ and $c+1$? Does it depend on what $a$, $b$ and $c$ are?