Problems

There are two piles of rocks, $10$ rocks in each pile. Fred and George play a game, taking the rocks away. They are allowed to take any number of rocks only from one pile per turn. The one who has nothing to take loses. If Fred starts, who has the winning strategy?

A group of $15$ elves decided to pay a visit to their relatives in a distant village. They have a horse carriage that fits only $5$ elves. In how many ways can they assemble the ambassador team, if at least one person in the team needs to be able to operate the carriage, and only $5$ elves in the group can do that?

There are $5$ pirates and they want to share $8$ identical gold coins. In how many ways can they do it if each pirate has to get at least one coin?

Is $100!=100\times 99\times ...\times 2$ divisible by $2^{100}$?

Prove the magic trick for the number $1089={33}^2$: if you take any $3$-digit number $\overline{abc}$ with digits coming in strictly descending order and subtract from it the number obtained by reversing the digits of the original number $\overline{abc}-\overline{cba}$ you get another $3$-digit number, call it $\overline{xyz}$. Then, no matter which number you started with, the sum $\overline{xyz}+\overline{zyx}=1089$.
Recall that a number $\overline{abc}$ is divisible by $11$ if and only if $a-b+c$ also is.

We want to wrap $12$ Christmas presents in different coloured paper. We have $6$ different patterns of paper and we want to use each one exactly twice. In how many ways can we do this?

Mr Roberts wants to place his little stone sculptures of vegetables on the different shelves around the house. He has $17$ sculptures in total and three shelves that can fit $7$, $8$ and $2$ sculptures respectively. In how many ways can he do this?
The order of sculptures on the shelf does not matter.

Do there exist two numbers such that their sum, quotient and product would be all equal to each other?

It is easy to construct one equilateral triangle using three identical matches. Is it possible to construct four equilateral triangles by adding just three more matches identical to the original ones?

Find the largest possible number of bishops that can be placed on the $8\times 8$ chessboard so that no two bishops threaten each other.