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. Prove that if \(n^2\) is divisible by \(2\), then \(n\) 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 \(x+7 \geq y+8\). Prove that \(x>y\).

There exist various ways to prove mathematical statements, one of the possible methods, which might come handy in certain situations is called Proof by contradiction. To prove a statement we first assume that the statement is false and then deduce something that contradicts either the condition, or the assumption itself, or just common sense. Thereby concluding that the first assumption must have been wrong, so the statement is actually true.

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 into red and yellow such that there are the same number of red cells and yellow cells?