Problems

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 $2$.

Let $n$ be an integer. Prove that if $n^3$ is divisible by $3$, then $n$ is divisible by $3$.

The numbers $x$ and $y$ satisfy $x+3=y+5$. Prove that $x>y$.

The numbers $x$ and $y$ satisfy $x+7\ge y+8$. Prove that $x>y$.

Can three points with integer coordinates be the vertices of an equilateral triangle?

Prove that there are infinitely many natural numbers $\{1,2,3,4,...\}$.

Prove that there are infinitely many prime numbers $\{2,3,5,7,11,13...\}$.

Is it possible to colour the cells of a $3\times 3$ board red and yellow such that there are the same number of red cells and yellow cells?

Prove the divisibility rule for $25$: a number is divisible by $25$ if and only if the number made by the last two digits of the original number is divisible by $25$;
Can you come up with a divisibility rule for $125$?