AM-GM Inequality (AIME / USAJMO / AMO)

Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.

Proof track: treat each problem as a full olympiad writeup on paper. Here you only enter the final answer (integer or short expression).

Easy Medium Hard AIME / USAJMO / AMO

AM-GM Inequality: core idea

Proofs and applications.

Example problem: hint and answer

**(Proof track.)** Prove AM-GM for $n$ positive reals by induction (or Jensen). Equality holds iff all equal. Enter the maximum of $xyz$ given $x+y+z=12$ and $x,y,z>0$ .

Hint: Equality at $x=y=z=4$ . Answer: 64.

Practice (9 problems — type your answers)

No proof submission required — answers only.

1. **(Proof track.)** Prove Cauchy–Schwarz $(\sum a_i^2)(\sum b_i^2)\ge(\sum a_ib_i)^2$ . Enter the minimal value of $x^2+y^2+z^2$ given $x+y+z=6$ .
2. **(Proof track.)** Prove weighted AM-GM and find $\min x^2y$ for $2x+y=6$ , $x,y>0$ . Enter the minimum.
3. **(Proof track.)** Prove Nesbitt’s inequality $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\ge\frac32$ for positives. Enter the value of the left side when $a=b=c=1$ .
4. **(Proof track.)** Prove by telescoping that $\prod_{k=1}^{n}\big(1+\frac1k\big)=n+1$ . Enter the product for $n=5$ .
5. **(Proof track.)** Prove AM-GM for $n$ positive reals by induction (or Jensen). Equality holds iff all equal. Enter the maximum of $xyz$ given $x+y+z=12$ and $x,y,z>0$ .
6. **(Proof track.)** Prove Cauchy–Schwarz $(\sum a_i^2)(\sum b_i^2)\ge(\sum a_ib_i)^2$ . Enter the minimal value of $x^2+y^2+z^2$ given $x+y+z=6$ .
7. **(Proof track.)** Prove weighted AM-GM and find $\min x^2y$ for $2x+y=6$ , $x,y>0$ . Enter the minimum.
8. **(Proof track.)** Prove Nesbitt’s inequality $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\ge\frac32$ for positives. Enter the value of the left side when $a=b=c=1$ .
9. **(Proof track.)** Prove by telescoping that $\prod_{k=1}^{n}\big(1+\frac1k\big)=n+1$ . Enter the product for $n=5$ .