Let \(CB\) and \(CD\) be tangents to the circle with the centre \(A\), let \(E\) be the point of intersection of the line \(AC\) with the circle. Draw \(FG\) as the segment of a tangent drawn through the point \(E\) between the lines \(CB\) and \(CD\). Find \(FG\) if the radius of the circle is \(15\) and \(AC = 39\).
Prove that the relation between areas of two similar polygons equals to the square of their similarity coefficient.
In the triangle \(ABC\) with a right angle \(\angle ACB\), \(CD\) is the height and \(CE\) is the bisector. Draw the bisectors \(DF\) and \(DG\) of the triangles \(BDC\) and \(ADC\). Prove that \(CFEG\) is a square.
Karl and Louie are playing the following game. There is a round table that has \(24\) seats around it. Karl and Louie place action figures around the table. However, no two figures are allowed to sit next to each other, regardless if they belong to Karl or Louie. The player, who cannot place their figure loses the game, Karl goes first - show that Louie will always win.
Katie and Andy play the following game: There are \(18\) chocolate bites on a plate. Each player is allowed to take \(1,2\) or \(3\) bites at once. The person who cannot take any more bites loses. Katie starts. Who has the winning strategy?
Arthur and Dan play the following game. There are \(26\) beads on the necklace. Each boy is allowed to take \(1,2,3\) or \(4\) beads at once. The boy who cannot take any more beads loses. Arthur starts - who will win?
Two goblins, Krok and Grok, are playing a game with a pile of gold. Each goblin can take any positive number of coins no larger than \(9\) from the pile. They take moves one after another. There are \(3333\) coins in total, the goblin who takes the last coin wins. Who will win, if Krok goes first?
There are all the numbers from \(1\) to \(2020\) written on the board. Karen and Leon are playing a game where they pick a number off the board and wipe it, together with all of its divisors. Leon goes first - prove that Karen always loses.
Katie and Juan played chess for some time and they got bored - Katie was winning all the time. She decided to make the game easier for Juan and changed the rules a bit. Now, each player makes two usual chess moves at once, and then the other player does the same. (Rules for checks and check-mates are modified accordingly). In the new game, Juan will start first. Show that Katie definitely does not have a winning strategy.
Two players are emptying two drawers full of socks. One drawer has 20 socks and the other has 34 socks. Each player can take any number of socks from one drawer. A player who can’t make a move loses. Who will win, the first or the second player?