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AM-GM Inequality (Easy)

AMC 10 Q1–8 depth — contest entry, not school drills.

Easy Medium Hard AIME / USAJMO / AMO

AM-GM Inequality: core idea

Two terms.

Example problem: hint and answer

Solve $x+3>7$ for the least integer $x$ .

Hint: Isolate the variable; reverse the sign when multiplying by a negative. Answer: 5.

Practice (9 problems — type your answers)

1. Solve $2x\le10$ for the greatest integer $x$ .
2. Find the minimum of $x^2$ for real $x$ .
3. If $x>4$ , is $x=5$ allowed? Type $1$ for yes.
4. Solve $-x<3$ for the least integer $x$ .
5. For positive $a,b$ with $a+b=10$ , find the maximum possible $ab$ .
6. Find the minimum of $(x-2)^2$ .
7. Solve $3x+1\ge10$ for the least integer $x$ .
8. For positive $x$ , find the minimum of $x+1/x$ .
9. Solve $5-x\ge2$ for the greatest integer $x$ .

AM-GM Inequality Practice Problems — Easy