Problems

Prove that $n^3-n$ is divisible by $24$ for any odd $n$.

Show that if numbers $a-b$ and $c-d$ are divisible by $11$, then $ac-bd$ and $ad-bc$ are also both divisible by $11$.

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 $y$.

Inside a square with side 1 there are several circles, the sum of the radii of which is 0.51. Prove that there is a line that is parallel to one side of the square and that intersects at least 2 circles.

Solve the equation $\lfloor x^3\rfloor +\lfloor x^2\rfloor +\lfloor x\rfloor =\{x\}-1$.

In the US, it is customary to record the date as follows: the number of the month, then the number of the day and then the year. In Europe, the number comes first, then the month and then the year. How many days are there in the year, the date of which can be read definitively, without knowing how it was written?

The cells of a $15\times 15$ square table are painted red, blue and green. Prove that there are two lines which at least have the same number of cells of one colour.

What is the maximum number of kings, that cannot capture each other, which can be placed on a chessboard of size $8\times 8$ cells?