Problems

Each cell of a \(3 \times 3\) square can be painted either black, or white, or grey. How many different ways are there to colour in this table?

A circular triangle is a triangle in which the sides are arcs of circles. Below is a circular triangle in which the sides are arcs of circles centered at the vertices opposite to the sides.

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 the circular triangle.

In a box there are \(20\) cards of blue, red, and yellow colours. Yellow cards outnumber red ones and there are six times less blue cards than yellow ones. What is the minimum number of cards that need to be drawn from the box without looking, to guarantee a red card among them?

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 \(2^{100}\)?

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.