Problems

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 the length $|FG|$ 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 triangle $\mathrm{△}ABC$ with right angle $\mathrm{\angle }ACB={90}^{\circ }$, $CD$ 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 explore some mathematical games involving two players who move alternately. In many games, one of the players has a winning strategy, which guarantees victory regardless of the opponent’s moves. We now describe a systematic approach that can be very helpful, though it is not needed all the time.

Let’s say you are at a stage of the game where you can win in one move and it is also your turn. Then that position is called a winning position. We can now make the following definition for all states of the game.

A losing position is one where all your moves give the other player a winning position. A winning position is one where you can make a move that gives your opponent a losing position.

We can analyze from the end of the game backward and figure out whether each possible state is a winning position or a losing position. The first player has a winning strategy if the starting position is a winning position. The winning strategy belongs to the second player if the starting position is a losing position.

Karl and Louie are playing a game. They place action figures around a round table with 24 seats. No two figures are allowed to sit next to each other, regardless of whether they belong to Karl or Louie. The player who cannot place their figure loses the game. Karl goes first - show that Louie can 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?

The numbers from $1$ to $2025$ are written on a board. Karen and Leon are playing a game where they pick a number on the board and wipe it, together with all of its divisors. Leon goes first. Show that he has a winning strategy.

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.