Problems

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 \(2^{100}\)?

It is known that \(a + b + c = 5\) and \(ab + bc + ac = 5\). What are the possible values of \(a^2 + b^2 + c^2\)?

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.

On the diagram below \(AD\) is the bisector of the triangle \(ABC\). The point \(E\) lies on the side \(AB\), with \(AE = ED\). Prove that the lines \(AC\) and \(DE\) are parallel.

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?

Winnie the Pooh has five friends, each of whom has pots of honey in their house: Tigger has \(1\) pot, Piglet has \(2\), Owl has \(3\), Eeyore has \(4\), and Rabbit has \(5\). Winnie the Pooh comes to visit each friend in turn, eats one pot of honey and takes the other pots with him. He came into the last house carrying \(10\) pots of honey. Whose house could Pooh have visited last?

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.

Does there exist a power of \(3\) that ends in \(0001\)?