Algebra
# Classical Inequalities

$A = \frac{a+b}{2}$ and the geometric mean $G = \sqrt{ab}$ of non-negative quantities $a$ and $b$ are shown. Based on the diagram, which inequality is correct?

The arithmetic meanIf the sum of two positive numbers is 18, what is the largest possible value of their product?

*Hint.* For non-negative numbers $a$ and $b,$ the Arithmetic Mean - Geometric Mean Inequality implies that $\frac{a+b}{2} \geq \sqrt{ab},$ and $\frac{a+b}{2} = \sqrt{ab} \text{ when } a = b.$

For values of $x$ such that $(8 +x)$ and $(1 -x)$ are positive, what is the maximum possible value of

$(8 +x)(1-x)?$

*Hint.* Apply the Arithmetic Mean - Geometric Mean Inequality.

If the sum of two legs of a right triangle is 20, what is the smallest possible value of the hypotenuse?

**Hint.** The *quadratic mean*, or *root-mean-square* of non-negative numbers $a$ and $b$ is greater than or equal to their arithmetic mean, i.e.

$\sqrt{\frac{a^2+b^2}{2}} \geq \frac{a+b}{2},$

with equality happening when $a = b.$

What is the quadratic mean of the four values $0, 0, 0,$ and $8?$

**Note.** The *quadratic mean*, or *root-mean-square*, of non-negative numbers $x_1, \ldots, x_n$ is

$\sqrt{\frac{x_1^2+\cdots + x_n^2}{n}}.$