TPA's equality

TPA's equality is an inequality discovered by AoPS user thepiercingarrow in 2017, said to have been inspired from doing calculus with snek.

Inequality

Let $F$ be any symmetric inequality. Let $x_1, \cdots,x_n$ be the parts of that inequality. Then in contest space, $F$ achieves equality when $x_1 = \cdots = x_n.$

Proof

The proof of TPA's equality is very complicated. It requires using quantum computing to search all inequalities over contest space and checking the verity of the TPA equality for each one. A program was developed by alifenix- in TPA-Snek Labs in Equestria, and ran on the TPA-Snek quantum computer. This program is now open-sourced and available for anyone to run. It can be found on the TPA-Snek Labs main website.

Problems

Introductory

Prove the equality case of AM-GM using TPA's equality.

Intermediate

  • Prove that for any $\triangle ABC$, we have the maximum of $\sin{A}+\sin{B}+\sin{C}$ is $\frac{3\sqrt{3}}{2}$.

Olympiad

  • Let $a,b,c$ be positive real numbers. Prove that the minimum of $\frac{a}{\sqrt{a^{2}+8bc}}+\frac{b}{\sqrt{b^{2}+8ca}}+\frac{c}{\sqrt{c^{2}+8ab}}$ is $1$ (Source)
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