Problems

The numbers from \(1\) to \(9\) are written in a row. Is it possible to write down the same numbers from \(1\) to \(9\) in a second row beneath the first row so that the sum of the two numbers in each column is an exact square?

On a Halloween night ten children with candy were standing in a row. In total, the girls and boys had equal amounts of candy. Each child gave one candy to each person on their right. After that, the girls had \(25\) more candy than they used to. How many girls are there in the row?

There are \(16\) cubes, each face of every cube is coloured yellow, black, or red (different cubes can be coloured differently). After looking at their colouring pattern, Pinoccio said that he could put all the cubes on the table in such a way that only the yellow color would be visible, on the next turn he could put the cubes in such a way that only the black color would be visible, and also he could put them in such a way that only the red color would be visible. Is there a colouring of the cubes such that he could tell the truth?

Alex writes natural numbers in a row: \(123456789101112...\) Counting from the beginning, in what places do the digits \(555\) first appear? For example, \(101\) first appears in the 10th, 11th and 12th places.

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 \(a,b,n\).