Welcome all to the first ever 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.
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.
Inequalities allowed to be used are AM-GM, Muirhead, Power Mean and Weighted Power Mean, Cauchy-Schwarz, Holder, Rearrangement, Chebyshev, Schur, Jensen, Karamata, Reverse Rearrangement 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. And above all else, have fun!!!
Let \(x\), \(y\) and \(z\) be positive reals such that \(x+y \geq z\), \(y+z \geq x\) and \(z+x \geq y\). Find families of solutions for \((x, y, z)\) such that the following inequality is satisfied.
\[2x^2 (y+z) + 2y^2 (z+x) + 2z^2 (x+y) \geq x^3 + y^3 + z^3 + 9xyz\]
P.S.: For those who want to discuss problem solutions, they can do so here.