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AM-GM Inequality (Medium)

AMC 10/12 mid (≈Q9–16) — multi-step.

Easy Medium Hard AIME / USAJMO / AMO

AM-GM Inequality: core idea

Equality cases.

Example problem: hint and answer

For positive $a,b$ with $a+b=12$ , find the maximum of $ab$ .

Hint: For AM-GM, equality gives the extremum when terms are equal. Answer: 36.

Practice (9 problems — type your answers)

1. Find the minimum of $x^2-6x+10$ .
2. For positive $x$ , find the minimum of $2x+8/x$ .
3. Solve $|x-3|<2$ . Find the greatest integer solution.
4. For positive $a,b$ with $ab=16$ , find the minimum of $a+b$ .
5. Find the maximum of $-x^2+8x-7$ .
6. For positive $x,y$ with $x+y=20$ , find the maximum of $xy$ .
7. Solve $x^2<16$ . Find the greatest integer $x$ .
8. For positive $x$ , find the minimum of $x^2+4/x^2$ .
9. If $a,b>0$ and $a+b=6$ , find the maximum of $\sqrt{ab}$ .

AM-GM Inequality Practice Problems — Medium