Problems

Ten players were entered into a badminton tournament. The first round consisted of 5 matches, with each player in one match. In how many different ways could the 10 players be matched against each other?

There are again some adventurers standing in a queue to see a dragon’s treasure. This time, there are more of them – $26$. The rules have changed slightly, they still enter exactly in the order they are queuing, but they now have to divide themselves into $5$ groups, and some of the groups can be empty, do not consist of any adventurers at all. In how many ways can they do that now?

Problems often involve a protagonist, a quest and a story. In combinatorics, stories can help us prove identities and formulas, that would be difficult to prove otherwise. Here, you can write your own story, which will show that the following statement is always true:

The number of ways we can choose $k$ out of $n$ items is equal to the number of ways we can choose $k$ out of $n-1$ objects PLUS the number of ways in which we can choose $k-1$ out of $n-1$ objects.

In a trapezium $ABCD$, the side $AB$ is parallel to the side $CD$. Show that the areas of triangles $\mathrm{△}ABC$ and $\mathrm{△}ABD$ are equal.

The triangle visible in the picture is equilateral. The hexagon inside is a regular hexagon. If the area of the whole big triangle is $18$, find the area of the small blue triangle.

On the left there is a circle inscribed in a square of side 1. On the right there are 16 smaller, identical circles, which all together fit inside a square of side 1. Which area is greater, the yellow or the blue one?

In a pentagon $ABCDE$, diagonal $AD$ is parallel to the side $BC$ and the diagonal $CE$ is parallel to the side $AB$. Show that the areas of the triangles $\mathrm{△}ABE$ and $\mathrm{△}BCD$ are the same.

Which triangle has the largest area? The dots form a regular grid.

What is the ratio between the red and blue area? All shapes are semicircles.

In a parallelogram $ABCD$, point $E$ belongs to the side $CD$ and point $F$ belongs to the side $BC$. Show that the total red area is the same as the total blue area: