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Tommy and Claire are going to get some number of game tokens tomorrow. They are planning to play a game: each player can take \(1,4\) or \(5\) tokens from the total. The person who can’t take any more loses. Claire will start. They don’t know how many tokens they will get. They might get a number between \(1\) and \(2020\). In how many cases Claire will win?

Fred and Johnny have a number \(1000\) written on the board. In their turn, a player wipes out a number currently on the board and replaces it with either a number \(1\) smaller, or half of the number on the board (rounded down). A player that writes \(0\) on the board wins. Johnny starts, who will win?

You take nine cards out of a standard deck (ace through 9 of hearts), put them all face up on a table and play the following game against another player:
Both players take turns choosing a card. The first player to have three cards that add up to 15 wins. The ace counts as one.
If both players play optimally, which player has a winning strategy?

Andy and Melissa are playing a game using a rectangular chocolate bar made of identical square pieces arranged in \(50\) rows and \(20\) columns. A move is to divide the bar into two parts along the division line. Two parts of the bar stay in the game as separate pieces and cannot be rotated, but both can continue to be divided. However, Melissa can only cut along the vertical lines and Andy can only cut along the horizontal lines. Melissa starts. Who will win?

Terry and Janet are playing a game with stones. There are two piles of stones, one has \(m\) stones and the other has \(n\) stones initially. In their turn, a player takes from one pile a positive number of stones that is a multiple of the number of stones in the other pile at that moment. The player who cleans up one of the piles wins. Terry starts - who will win?

Let’s look at triangular numbers, numbers which are a sum of the first \(n\) natural numbers: \[1+2+3+\dots +n\] Show using induction that the \(n\)-th triangular number is equal to \(\frac{n(n+1)}2\).

Show using induction that \[1+3+5+\dots+ (2n-1) = n^2\] The sum of \(n\) first odd numbers is equal to \(n^2\).

Two convex polygons \(A_1A_2...A_n\) and \(B_1B_2...B_n\) have equal corresponding sides \(A_1A_2 = B_1B_2\), \(A_2A_3 = B_2B_3\), ... \(A_nA_1 = B_nB_1\). It is also known that \(n - 3\) angles of one polygons are equal to the corresponding angles of the other. Prove that the polygons \(A_1...A_n\) and \(B_1...B_n\) are equal.

Show that \(2^{2n} - 1\) is always divisible by \(3\), if \(n\) is a positive natural number.

The famous Fibonacci sequence is a sequence of numbers, which starts from two ones, and then each consecutive term is a sum of the previous two. It describes many things in nature. In a symbolic form we can write: \(F_0 = 1, F_1 = 1, F_n = F_{n-1} + F_{n-2}\).
Show that \[F_0+F_1+ F_2 + \dots + F_n = F_{n+2}-1\]