Problems

Which of the two fractions is larger? \[\frac{1\overbrace{00\cdots 00}^{1984\text{ zeroes}}1 }{1\underbrace{00\cdots 000}_{1985\text{ zeroes}}1}\qquad \text{or}\qquad \frac{1\overbrace{00\cdots 00}^{1985\text{ zeroes}}1 }{1\underbrace{00\cdots 000}_{1986\text{ zeroes}}1}\]

Which is larger? \[95^2+96^2\qquad \text{or}\qquad 2\times 95\times 96\]

Among all rectangles with perimeter \(4\), show that the one with largest area is a square, and determine that largest area.

Show how the following diagram “proves" that \((a+b)^2=a^2+2ab+b^2\) without just expanding the brackets:

Show that the consecutive sum of odd numbers from \(1\) until any odd number is a perfect square. For example: \(1+3+5=3^2\), or \(1+3+5+7+9=5^2\).

Explain how you can use the diagram below to show that \(a^2-b^2=(a+b)(a-b)\)

The Arithmetic-Geometric inequality is one of the most famous inequalities. It says that for positive numbers \(a\) and \(b\), \(\frac{a+b}2\geq \sqrt{ab}\). Show this inequality using the diagram below:

In the examples we showed that the sum of consecutive odd numbers starting from one was a perfect square. Now show how the following diagram can be used to give an alternative proof.

Using the following diagram, show that \(1^3+2^3+3^3+\cdots+n^3=\frac{1}{4}\left(n(n+1)\right)^2\)

Three clubs take part in a festival. Each club has at least one member.

During the festival, every member of one club shakes hands with every member of another club. In total (counting all three pairs of clubs), there were \(75\) handshakes between people from different clubs.

What is the smallest possible total number of participants?