Problems

Tom found a large, old clock face and put 12 sweets on the number 12. Then he started to play a game with himself. In each move he moves one sweet to the next number clockwise, and some other to the next number anticlockwise. Is it possible that after finite number of steps there is exactly 1 of the sweets on each number?

Can you cover a $13\times 13$ square using two types of blocks: $2\times 2$ squares and $3\times 3$ squares?

What time is it going to be in $2019$ hours from now?

What is the remainder of $1203\times 1203-1202\times 1205$ when divided by $12$?

Show that a perfect square can only have remainders 0 or 1 when divided by 4.

Convert 2000 seconds into minutes and seconds.

What is a remainder of $7780\times 7781\times 7782\times 7783$ when divided by $7$?

Tim had more hazelnuts than Tom. If Tim gave Tom as many hazelnuts as Tom already had, then Tim and Tom would have the same number of hazelnuts. Instead, Tim gave Tom only a few hazelnuts (at most five) and divided his remaining hazelnuts equally between $3$ squirrels. How many hazelnuts did Tim give to Tom?

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 $11$.