Problems

Two circles are tangent to each other and the smaller circle with the center \(A\) is located inside the larger circle with the center \(C\). The radii \(CD\) and \(CE\) are tangent to the smaller circle and the angle \(\angle DCE = 60^{\circ}\). Find the ratio of the radii of the circles.

For positive real numbers \(a,b,c\) prove the inequality: \[(a^2b + b^2c + c^2a)(ab^2 + bc^2 + ca^2)\geq 9a^2b^2c^2.\]

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?

Let \(p\) and \(q\) be two prime numbers such that \(q = p + 2\). Prove that \(p^q + q^p\) is divisible by \(p + q\).

Consider a set of natural numbers \(A\), consisting of all numbers divisible by \(6\), let \(B\) be the set of all natural numbers divisible by \(8\), and \(C\) be the set of all natural numbers divisible by \(12\). Describe the sets \(A\cup B\), \(A\cup B\cup C\), \(A\cap B\cap C\), \(A-(B\cap C)\).

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.