Problems

Which number is larger \(1234567\times 1234569\) or \(1234568^2\)?

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 \(1^3+2^3+3^3+\cdots+n^3=\frac{1}{4}\left(n(n+1)\right)^2\)