Problems

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 a 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?

Mathematical Induction is a method to prove statements that are usually true for all natural numbers. The method consists of two steps.

• The first step, known as the base case, is to prove the given statement for the first natural number.

• The second step, known as the inductive step, is to prove that the true statement for the number $n$ implies that the statement for $n+1$ is also true.

To understand how the method of induction works we look at dominoes. Have you ever seen a line of dominoes falling? How does it happen?

To prove that a line of dominoes will all fall when we push the first one, we just have to prove that:

• The first domino falls down (base case)

• The dominoes are close enough that each domino will knock over the next one when it falls (inductive step).

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

Show using induction that $1+3+5+\cdots +(2n-1)=n^2$. That is, the sum of $n$ first odd numbers is equal to $n^2$.

Two convex polygons $A_1{A}_2...A_n$ and $B_1{B}_2...B_n$ have equal corresponding sides $A_1{A}_2=B_1{B}_2$, $A_2{A}_3=B_2{B}_3$, ... $A_n{A}_1=B_n{B}_1$. 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 congruent.

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 $$F_0+F_1+F_2+\cdots +F_n=F_{n+2}-1$$

In a certain country, there are $n$ cities. Some of them are connected by roads, all of which go in both directions. It is possible to get from any city to any other city using only roads. However, for any pair of cities, there is always only one way to get from one of them to the other and there are no alternative routes.
Show that there are exactly $n-1$ roads in this country.