Problems

The entire plane is coloured using two colours: red and blue. Prove that there must exist two points of the same colour that are exactly \(1\) meter apart.

Give a visual proof that the sum of consecutive numbers until \(n\), i.e: \(1+2+\cdots + n\), where \(n\) is some whole number; is equal to \(n(n+1)/2\).

Use a visual proof to find the value of \[\frac{1+3+5+\cdots +2n-1}{(2n+1)+(2n+3)+\cdots + (4n-1)}\] You are not allowed to use the result from the examples to simplify the fraction.

The Pythagorean Theorem is a very useful tool in geometry. It says that if you have a right-angled triangle with sides measuring \(a,b,c\) where \(c\) is the longest side of the three, then \(a^2+b^2=c^2\). Explain how the following diagram gives a visual proof of the Pythagoraen Theorem.