Algebra
# Classical Inequalities

If 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

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

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