Prove the \(GM-HM\) inequality for positive real numbers \(a_1\), \(a_2\), ..., \(a_n\): \[\sqrt[n]{a_1a_2...a_n} \geq \frac{n}{\frac{1}{a_1} + ... \frac{1}{a_n}}.\]
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.
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.