Problems

On a $10\times 10$ board, a bacterium sits in one of the cells. In one move, the bacterium shifts to a cell adjacent to the side (i.e. not diagonal) and divides into two bacteria (both remain in the same new cell). Then, again, one of the bacteria sitting on the board shifts to a new adjacent cell, either horizontally or vertically, and divides into two, and so on. Is it possible for there to be an equal number of bacteria in all cells after several such moves?

The distance between two villages equals $999$ kilometres. When you go from one village to the other, every kilometre you see a sign on the road, saying $0\mid 999,1\mid 998,2\mid 997,...,999\mid 0$. The signs show the distances to the two villages. Find the number of signs that contain only two different digits. For example, the sign $0\mid 999$ contains only two digits, namely $0$ and $9$, whereas the sign $1\mid 998$ contains three digits, namely $1$, $8$ and $9$.

Red, blue and green chameleons live on an island. One day $35$ chameleons stood in a circle. A minute later, they all changed colour at the same time, each changing into the colour of one of their neighbours. A minute later, everyone again changed their colours at the same time into the colour of one of their neighbours. Is it ever possible that each chameleon was each of the colours red, blue and green at some point? For example, it’s allowed for a chameleon to start off blue, turn green after one minute, then turn red after the second minute. It’s not allowed for a chameleon to start off blue, turn green after one minute, but then turn back to blue after the second minute.

The pigeonhole principle is often called “Dirichlet’s box principle". Dirichlet made good use of this tool to show a fundamental result in Diophantine approximation, now commonly known as the Dirichlet Approximation Theorem. You will now prove it yourself!

Suppose $\alpha$ is any irrational real number and $N\ge 1$ is any positive integer. Show that there is an integer $1\le q\le N$ and an integer $p$ such that $$|q\alpha -p|<\frac{1}{N}.$$

Consider a line segment of length $3m$. Jack chose $4$ random points on the segment and measured all the distances between those $4$ points. Prove that at least one of the distances is less than or equal to $1m$.

The kingdom of Triangland is an equilateral triangle of side $10$ km. There are $5$ cities in this kingdom. Show that some two of them are closer than $5$ km apart.

Margaret marked three points with integer coordinates on a number line with a red crayon. Meanwhile Angelina marked the midpoint of each pair of red points with a blue crayon. Prove that at least one of the blue points has an integer coordinate.

Margaret and Angelina coloured points in the second dimension. Now Margaret marked five points with both integer coordinates on a plane with a red crayon, while Angelina marked the midpoint for each pair of red points with a blue crayon. Prove that at least one of the blue points has both integer coordinates.

Anna has a garden of square shape with side $4$ m. After playing with her dog in the garden she left $5$ dog toys on the lawn. Show that some two of them are closer than $3$ m apart.

Twelve lines are drawn on the plane, passing through a point $A$. Prove that there are two of them with angle less than ${17}^{\circ }$ between them.