Problems

The area of the red triangle is \(25\) and the area of the orange triangle is \(49\). What is the area of the trapezium \(ABCD\)?

Prove that the ratio of perimeters of similar polygons is equal to the similarity coefficient.

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.

Today we will have a look on some mathematical games. In all the games today, there are two players, who make moves alternately. There is a certain goal in each of these games, common for both players. The player who achieves it, wins, and the game will end at some point, draws are not allowed. It turns out in such games one of the players has a winning strategy - no matter what the other one does, the player following the strategy will always win. Today we can look into finding who has the winning strategy and what it might be.

One important tool we have to investigate these games are winning and losing positions. If you playing the game can win in one go starting from a certain state of the board, number of tokens on the table, set of cards etc, or whatever describes the current state of the game, it is said you are in a winning position. However, if all the moves you can make give the winning position to your opponent, it is said you are in a losing position. You must do a move that will guarantee you opponent to win - or at least give them a certain opportunity if they are smart enough to take it.

We can go further and say that a winning position is such a position that you can always make a move that will change the state of the game to the losing position for your opponent. Whereas a losing position is such a position that you must make a move that handles the winning position to the other player.

In some simple games the strategy can be guessed without going into details about winning or losing positions. But often you can start from the last stages of the game, find out what positions are immediately winning, then find all the losing positions that lead only there. In this way, working your way from the end of the game towards its beginning, one can characterize all the possible positions as winning or losing. The player that starts the game in a winning position has a winning strategy - if they start the game in a losing position, the strategy belongs to their opponent.

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?