Welcome all to Season 2 of the Brilliant Inequality Contest. Like the Brilliant Integration Contest, the aim of the Inequality Contest is to improve skills and techniques often used in Olympiad-style Inequality problems. But above all, the main reason is so we can have fun.
Anyone is allowed to participate, as long as they adhere to the following rules.
I will post the first problem. If someone solves it, he or she can post a solution and then must post a new problem.
A solution must be posted below the thread of the problem. Then, the solver must post a new problem as a separate thread.
Please make a substantial comment. Spam or unrelated comments will be deleted. (To help me out, try to delete your comments within an hour of when you posted them if they are not the solutions.)
Make sure you know how to solve your own problem before posting it, in case no one else is able to solve it within 48 hours. Then, you must post the solution and you have the right to post a new problem.
If the one who solves the last problem does not post a new problem in 24 hours, the creator of the previous problem has the right to post another problem.
The scope of the problems is Olympiad-style inequalities, but non-standard inequalities are also allowed, (due to increasing number of non-standard inequalities found in olympiads)
You are not allowed to post problems requiring calculus in the solutions (use of differentiation to prove a curve is concave or convex is allowed).
Lagrange Multipliers are not allowed to be used in a solution.
Most Standard Inequalities are allowed to be used, including AM-GM, Muirhead, Power Mean and Weighted Power Mean, Cauchy-Schwarz, Holder, Rearrangement, Chebyshev, Schur, Jensen, Karamata, Reverse Rearrangement, Bernoulli's Inequality and Titu's lemma.
Try to post the simplest solution possible. For example, if someone posted a solution using Holder, Titu's and Cauchy, when there is a solution using only AM-GM, the latter is preferred.
Format your proof as follows:
SOLUTION OF PROBLEM (insert problem number here)
[Post your solution here]
PROBLEM (insert problem number here)
[Post your problem here]
Remember to reshare this note so it goes to everyone out there. Also remember to keep the comments sorted by Newest, so you can see the current problem. And above all else, have fun!!!
For real numbers , if for , then prove that
where is the average of .