Problems

In a distant village, there are $3$ houses and $3$ wells. Inhabitants of each house want to have access to all $3$ wells. Is it possible to build non-intersecting straight paths from each house to each well? All houses and well must be level (that is, none of them are higher up, like on a mountain, nor are any of them on lower ground, like in a valley).

In good conditions, bacteria in a Petri cup spread quite fast, doubling every second. If there was initially one bacterium, then in $32$ seconds the bacteria will cover the whole surface of the cup.

Now suppose that there are initially $4$ bacteria. At what time will the bacteria cover the surface of the cup?

A piece containing exactly $4$ black cells is cut out from a regular $8$ by $8$ chessboard. You are only allowed to cut along the edges of the cells and the piece must be connected - namely you cannot have cells attached only with a vertex, they have to share a common edge.

Find the largest possible area of such a piece.

A parliament has 650 members. In this parliament there is only one house and every member has at most three enemies. We wish to split this parliament into two separate houses in such a way that each member will have at most one enemy in the same house as them. We assume that hard feelings among members of parliament are mutual, namely if $A$ recognises $B$ as their enemy, then $B$ also recognises $A$ as their enemy.

Is this splitting possible?

Find all $n$ such that a closed system of $n$ gears in a plane can rotate. We call a system closed if the first gear wheel is connected to the second and the $n$th, the second is connected to the first and the third, the third is connected to the second and the fourth, the fourth is connected to the third and the fifth, and so on until the $n$th is connected to the $n-1$th and the first. In the picture, we have a closed system of three gears.

$ABC$ is a triangle. The circumscribed circle is the circle that touches all three vertices of the triangle $ABC$. It is also the smallest circle lying entirely outside the triangle. The center of the circumscribed circle is $D$.

The inscribed circle is the circle which touches all three sides of the triangle $ABC$. It is also the largest circle lying entirely inside the triangle. The center of the inscribed circle is $E$.

The points $D$ and $E$ are symmetric with respect to the segment $AC$. Find the angles of the triangle $ABC$.

How many subsets are there of $\{1,2,...,n\}$ (the integers from $1$ to $n$ inclusive) containing no consecutive digits? That is, we do count $\{1,3,6,8\}$ but do not count $\{1,3,6,7\}$.