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AM-GM Inequality (Hard)

Late AMC 10/12 through early AIME — dense computation.

Easy Medium Hard AIME / USAJMO / AMO

AM-GM Inequality: core idea

Weighted.

Example problem: hint and answer

For $x>0$ , find $\min(x+16/x)$ .

Hint: Apply AM-GM with equality conditions, or reduce to a balanced product. Answer: 8.

Practice (9 problems — type your answers)

1. Positive $a+b=20$ . Find $\max ab$ .
2. Positive $xy=36$ . Find $\min(x+y)$ .
3. For $x>0$ , find $\min(x^2+9/x^2)$ .
4. A rectangle has area $64$ . Find least possible perimeter.
5. Positive $abc=8$ . Find $\min(a+b+c)$ .
6. For $x>0$ , find $\min(2x+18/x)$ .
7. Positive $a+b=14$ . Find $\max a^2b^2$ .
8. For positive $x,y$ , find $\min(x/y+y/x)$ .
9. A square has perimeter $36$ . Find its area.

AM-GM Inequality Practice Problems — Hard