Problems

There are numbers \(1,2,3,4,5,6,7,8,9\) and \(10\) written on a board. Each time you make a "move" you can erase three of the remaining numbers, \(a,b\) and \(c\), and replace them with the numbers \(2a+b,2b+c\) and \(2c+a\). Is it possible to make all the \(10\) numbers left on the board equal?

On a certain island there are \(17\) grey, \(15\) brown and \(13\) crimson chameleons. If two chameleons of different colours meet, both of them change to the third colour. No other colour changes are allowed. Is it possible that after a few such colour transitions all the chameleons have the same colour?

Sixteen lightbulbs are arranged in a \(4 \times 4\) grid. Some of them are on, the other ones are off. You are allowed to change the state of all the bulbs in a column, in a row, or along any diagonal (note: there are \(14\) diagonals in total!). Is it possible to go from the arrangement on the left to the one on the right by repeating this operation?

The numbers \(1\) to \(2025\) are written on a board. In one move we can erase two numbers and replace them with the absolute value of their difference. Can we achieve a sequence consisting of only \(0\) after some number of moves?

Hard. Let \(\mathcal{S}\) be a finite set of at least two points in the plane. Assume that no three points of \(\mathcal S\) are collinear. A windmill is a process that starts with a line \(\ell\) going through a single point \(P \in \mathcal S\). The line rotates clockwise about the pivot \(P\) until the first time that the line meets some other point belonging to \(\mathcal S\). This point, \(Q\), takes over as the new pivot, and the line now rotates clockwise about \(Q\), until it next meets a point of \(\mathcal S\). This process continues indefinitely. Show that we can choose a point \(P\) in \(\mathcal S\) and a line \(\ell\) going through \(P\) such that the resulting windmill uses each point of \(\mathcal S\) as a pivot infinitely many times.

A magic square is a square filled with numbers, one in each cell, in such a way that the sums of the numbers in each row, each column and along each of the two main diagonals are the same. The value of this sum is known as the magic constant of the square. Show that in any \(4 \times 4\) magic square (which contains \(16\) numbers) the sum of all the numbers in the \(4\) central squares is also equal to the magic constant of the square.

There are \(12\) light bulbs placed on top of the numbers on a clock face. Initially, only the bulb at \(12\) is on. We can choose any \(6\) consecutive bulbs and change their state simultaneously, that is, if any were on, we turn them off, if any were off, we turn them on. Can we perform this operation multiple times so that in the end only the bulb at \(11\) is on?

A new customer comes to the hotel and wants a room. It happened today that all the rooms are occupied. What should you do?

Now imagine you got \(10\) new guests arriving to the completely full hotel. What should you do now?

The next day you have even harder situation: to the hotel, where all the rooms are occupied arrives a bus with infinitely many new customers. In the bus all the seats have numbers \(1,2,3...\) corresponding to all natural numbers. How to deal with this one?