Problems

Prove that the set of all finite subsets of natural numbers \(\mathbb{N}\) is countable. Then prove that the set of all subsets of natural numbers is not countable.

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, it doubles every second. If, initially there was one bacteria, then in \(32\) seconds they will cover the whole surface of the cup. In what time the bacteria will cover the surface of the cup, if initially there were \(4\) bacteria?

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 the mistake in the bakery the cake that was supposed to be shaped as two concentric pieces came out as the left side of picture. Find the smallest amount of pieces the cake should be cut in, to build the cake on the right side of 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 \(2024\)?