Problems

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. You can take it as a fact that the angles inside a triangle add up to \(180^\circ\).

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

Draw a table with \(n+1\) columns and \(n\) rows, such that each column contains the numbers \(1,2,3,\cdots, n\). Explain how this table can be used to give a visual proof of the following identity \[(1^1\times 1!)\times (2^2\times 2!)\times (3^3\times 3!)\times\cdots \times (n^n\times n!)=(n!)^{n+1}\]

By cleverly dividing a square of side length \(1\), show that the sum \[\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\cdots + \frac{1}{1024}= \frac{1023}{1024}\]

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 when the ordering does not matter. 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. \(|iz|=|z|\).

Consider a right-angled triangle and let \(\theta\) be one of its acute angles. We define the sine of \(\theta\), written \(\sin(\theta)\), as the length of the side opposite to \(\theta\) divided by the length of the hypotenuse. Similarly, we define the cosine of \(\theta\), written \(\cos(\theta)\), as the length of the side adjacent to \(\theta\) divided by the length of the hypotenuse.

Now take a right-angled triangle with acute angle \(\alpha\), and on its hypotenuse build another right-angled triangle with acute angle \(\beta\). Use the resulting diagram to show that \(\sin(\alpha+\beta)=\sin(\alpha)\cos(\beta)+\sin(\beta)\cos(\alpha)\).

Among \(12\) identical-looking balls, exactly one has a different weight (we do not know whether it is heavier or lighter than the others).

Using a balance scale, show how to determine the odd ball, and whether it is lighter or heavier, using only three weighings.

Three positive numbers \(a,b,c\) satisfy \(ac-bc+ab=63\). What is the smallest value that \(a^2+b^2+c^2\) can be?