Problems

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.

On the diagram below the line \(BD\) is the bisector of the angle \(\angle ABC\) in the triangle \(ABC\). A line through the vertex \(C\) parallel to the line \(BD\) intersects the continuation of the side \(AB\) at the point \(E\). Find the angles of the triangle \(BCE\) triangle if \(\angle ABC = 110^{\circ}\).

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.

In a certain state, there are three types of citizens:

• A fool considers everyone a fool and themselves smart;

• A modest clever person knows truth about everyone’s intellectual abilities and consider themselves a fool;

• A confident clever person knows about everyone intellectual abilities correctly and consider themselves smart.

There are \(200\) deputies in the High Government. The Prime Minister conducted an anonymous survey of High Government members, asking how many smart people are there in the High Government. After reading everyone’s response he could not find out the number of smart people. But then the only member who did not participate in the survey returned from the trip. They filled out a questionnaire about the entire Government including themselves and after reading it the Prime Minister understood everything. How many smart could there be in the High Government (including the traveller)?