Problems

Find the representation of \((a+b)^n\) as the sum of \(X_{n,k}a^kb^{n-k}\) for general \(n\). Here by \(X_{n,k}\) we denote coefficients that depend only on \(k\) and \(n\).

The positive real numbers \(a, b, c, x, y\) satisfy the following system of equations: \[\left\{ \begin{aligned} x^2 + xy + y^2 = a^2\\ y^2 + yz + z^2 = b^2\\ x^2 + xz + z^2 = c^2 \end{aligned} \right.\]

Find the value of \(xy + yz + xz\) in terms of \(a, b,\) and \(c.\)

Sometimes different areas in mathematics are more related than they seem to be. A lot of algebraic expressions have geometric interpretation, and a lot of them can be used to solve problems in number theory.

Today we will solve several logic problems that revolve about a very simple idea. Imagine you are in a room in a dungeon and you can see doors leading out of the room. Some of them lead to the treasure and some of them lead to traps. It is possible that all doors lead to treasure or all lead to traps, but it is also possible that one door leads to treasure and all other lead to traps. Unless specified, there is always something behind the door.
Each door has a sign with a statement on it, but those statements are not always true. You have a dungeon guide, who is always honest with you and will tell you something about the truthfulness of the statements on the doors, but it will be up to you to put it all together and pick the correct door... or walk away, if you believe there is no treasure.

This is a famous problem, called Monty Hall problem after a popular TV show in America.
In the problem, you are on a game show, being asked to choose between three doors. Behind each door, there is either a car or a goat. You choose a door. The host, Monty Hall, picks one of the other doors, which he knows has a goat behind it, and opens it, showing you the goat. (You know, by the rules of the game, that Monty will always reveal a goat.) Monty then asks whether you would like to switch your choice of door to the other remaining door. Assuming you prefer having a car more than having a goat, do you choose to switch or not to switch?

Find a representation as a product of \(a^{2n+1} + b^{2n+1}\) for general \(a,b,n\).

Each integer on the number line is coloured either white or black. The numbers \(2016\) and \(2017\) are coloured differently. Prove that there are three identically coloured integers which sum to zero.

There are \(100\) non-zero numbers written in a circle. Between every two adjacent numbers, their product was written, and the previous numbers were erased. It turned out that the number of positive numbers after the operation coincides with the amount of positive numbers before. What is the minimum number of positive numbers that could have been written initially?

Let \(r\) be a rational number and \(x\) be an irrational number (i.e. not a rational one). Prove that the number \(r+x\) is irrational.
If \(r\) and \(s\) are both irrational, then must \(r+s\) be irrational as well?

Definition: We call a number \(x\) rational if there exist two integers \(p\) and \(q\) such that \(x=\frac{p}{q}\). We assume that \(p\) and \(q\) are coprime.
Prove that \(\sqrt{2}\) is not rational.