Problems

Using the area of a rectangle prove that $a\times b=b\times a$.

Jason has $20$ red balls and $14$ bags to store them. Prove that there is a bag which contains at least two balls.

One of the most useful tools for proving mathematical statements is the Pigeonhole principle. Here is one example: suppose that a flock of $10$ pigeons flies into a set of $9$ pigeonholes to roost. Prove that at least one of these $9$ pigeonholes must have at least two pigeons in it.

Show the following: Pigeonhole principle strong form: Let $q_1,q_2,...,q_n$ be positive integers. If $q_1+q_2+...+q_n-n+1$ objects are put into $n$ boxes, then either the $1$st box contains at least $q_1$ objects, or the $2$nd box contains at least $q_2$ objects, . . ., or the $n$th box contains at least $q_n$ objects.
How can you deduce the usual Pigeonhole principle from this statement?

Each integer on the number line is coloured either white or black. The numbers $2016$ and $2017$ are coloured differently. Prove that there are three identically coloured integers which sum to zero.

Each integer on the number line is coloured either yellow or blue. Prove that there is a colour with the following property: For every natural number $k$, there are infinitely many numbers of this colour divisible by $k$.

There are $100$ non-zero numbers written in a circle. Between every two adjacent numbers, their product was written, and the previous numbers were erased. It turned out that the number of positive numbers after the operation coincides with the amount of positive numbers before. What is the minimum number of positive numbers that could have been written initially?

Let $r$ be a rational number and $x$ be an irrational number (i.e. not a rational one). Prove that the number $r+x$ is irrational.
If $r$ and $s$ are both irrational, then must $r+s$ be irrational as well?

Definition: We call a number $x$ rational if there exist two integers $p$ and $q$ such that $x=\frac{p}{q}$. We assume that $p$ and $q$ are coprime.
Prove that $\sqrt{2}$ is not rational.

Let $n$ be an integer such that $n^2$ is divisible by $2$. Prove that $n$ is divisible by $2$.