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.

Without carrying out the multiplications, which is larger: \[(2015+2026)^2\qquad \text{or} \qquad 4\times 2015\times 2016\]

Give a visual proof of the following identity \[(1^1\times 1!)\times (2^2\times 2!)\times (3^3\times 3!)\times\cdots \times (k^k\times k!)=(k!)^{k+1}\]

By cleverly dividing a square of side length \(1\), show that the sum \[\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots + \frac{1}{1024}= \frac{1023}{1024}\]

For natural numbers \(n\) and \(k\) with \(k\leq n\), the notation \({n\choose k}\) means the number of ways one can choose \(k\) objects from a set of \(n\) objects. Explain how the diagram below gives a visual proof of the fact that \[{n+1\choose 2}={n\choose 2}+n.\]

For a real number \(x\), we call \(|x|\) its absolute value. It is defined as whichever is larger: \(x\) or \(-x\). For example, \(|-2|=2\) and \(|3|=3\).

One of the most important inequalities involving absolute values is the triangle inequality, which states that \[|a+b| \le |a| + |b|.\]

Show that this inequality is true.

Let \(z\) be a complex number. Show that

1. For a real number \(k\), \(|kz|=|k|\cdot |z|\).

2. \(|iz|=|z|\).