- 17,507

- 7,077

Challenge Level: High School

**RULES:**

1) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored.

2) It is fine to use nontrivial results without proof as long as you cite them and as long as it is "common knowledge to all mathematicians". Whether the latter is satisfied will be decided on a case-by-case basis.

3) If you have seen the problem before and remember the solution, you cannot participate in the solution to that problem.

4) You are allowed to use google, wolframalpha or any other resource. However, you are not allowed to search the question directly. So if the question was to solve an integral, you are allowed to obtain numerical answers from software, you are allowed to search for useful integration techniques, but you cannot type in the integral in wolframalpha to see its solution.

**CHALLENGE:**

If

*a, b*and

*c*are positive real numbers show that ##a^{a}b^{b}c^{c} \geq (abc)^{\frac{a + b + c}{3}}##