Problems

Let \(C_1\) and \(C_2\) be two concentric circles with \(C_1\) inside \(C_2\) and the center \(A\). Let \(B\) and \(D\) be two points on \(C_1\) that are not diametrically opposite. Extend the segment \(BD\) past \(D\) until it meets the circle \(C_2\) in \(C\). The tangent to \(C_2\) at \(C\) and the tangent to \(C_1\) at \(B\) meet in a point \(E\). Draw from \(E\) the second tangent to \(C_2\) which meets \(C_2\) at the point \(F\). Show that \(BE\) bisects angle \(\angle FBC\).

In a distant village of \(3\) houses all villagers want to have access to all \(3\) wells. Is it possible to build non-intersecting straight paths from each house to each well?

In good conditions, bacteria in a Petri cup spreads quite fast, doubling every second. If there was initially one bacteria, 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?

Find the area of the largest piece cut out from a regular chessboard, which contains exactly \(4\) black cells. 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, there has to be a common edge.

Due to a mistake in the bakery, a cake that was supposed to be shaped as two concentric pieces (like on the right diagram below) came out as like the left diagram below. Find the smallest number of pieces the cake should be cut into in order to rearrange the pieces into the cake on the right side of the picture.

In a parliament with only one house every member had not more than three enemies. Is it possible split this parliament into two separate houses in such a way that each member will have not more than 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.

Find all such \(n\), that the closed system of \(n\) gears on a plane can move. 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, and so on. On the picture we have a closed system of three gears.

Find the angles of the triangle \(ABC\) if the center of the inscribed circle \(E\) and the center of the superscribed circle \(D\) are symmetric with respect to the segment \(AC\).

Katie and Charlotte had \(4\) sheets of paper. They decided to cut some of the sheets into \(4\) pieces, then, some of the newly obtained papersheets they also cut into \(4\). In the end they counted the number of all sheets. Could this number be \(2024\)?

In a scout group among any four participants there is at least one, who knows three other. Prove that there is at least one participant, who knows the rest of the group.