Difference between revisions of "TPA's equality"

(Go ask tpa for clarification before labelling it as "fake)
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'''TPA's equality''', also known as the TPA-snek equality, is an equality discovered by AoPS user thepiercingarrow in 2017, said to have been inspired from doing calculus with snek.
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'''TPA's equality''' is an inequality discovered by AoPS user thepiercingarrow in 2017, said to have been inspired from doing calculus with snek.
 
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==Inequality==
     
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Let <math>F</math> be any symmetric inequality. Let <math>x_1, \cdots,x_n</math> be the parts of that inequality. Then in contest space, <math>F</math> achieves equality when <math>x_1 = \cdots = x_n.</math>
− '''DISCLAIMER: THIS IS DEFINITELY A REAL EQUALITY AND YOU CAN TOTALLY USE THIS ON THE AMO NEXT YEAR'''
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==Proof==
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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.
 
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==Inequality==
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==Problems==
 
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===Introductory===
Let <math>F</math> be any symmetric inequality involving positive numbers. Let <math>x_1, \cdots,x_n</math> be the parts of that inequality. In contest space, if <math>F</math> achieves the equality then it is for <math>x_1 = \cdots = x_n.</math>
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Prove the equality case of AM-GM using TPA's equality.
 
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===Intermediate===
 
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* Prove that for any <math>\triangle ABC</math>, we have the maximum of <math>\sin{A}+\sin{B}+\sin{C}</math> is <math>\frac{3\sqrt{3}}{2}</math>.
==Proof==
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===Olympiad===
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.
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*Let <math>a,b,c</math> be positive real numbers. Prove that the minimum of <math>\frac{a}{\sqrt{a^{2}+8bc}}+\frac{b}{\sqrt{b^{2}+8ca}}+\frac{c}{\sqrt{c^{2}+8ab}}</math> is <math>1</math> ([[2001 IMO Problems/Problem 2|Source]])
 
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[[Category:Inequality]]
 
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[[Category:Theorems]]
==Problems==
 
 
 
 
 
 
===Introductory===
 
 
 
Prove the equality case of AM-GM using TPA's equality.
 
 
 
 
 
 
===Intermediate===
 
 
 
* Prove that for any <math>\triangle ABC</math>, we have the maximum of <math>\sin{A}+\sin{B}+\sin{C}</math> is <math>\frac{3\sqrt{3}}{2}</math>.
 
 
 
 
 
 
===Olympiad===
 
 
 
*Let <math>a,b,c</math> be positive real numbers. Prove that the minimum of <math>\frac{a}{\sqrt{a^{2}+8bc}}+\frac{b}{\sqrt{b^{2}+8ca}}+\frac{c}{\sqrt{c^{2}+8ab}}</math> is <math>1</math> ([[2001 IMO Problems/Problem 2|Source]])
 
 
 
 
 
 
− ==Extension to all reals==
 
 
 
− *Consider <math>x^4+y^4 - 2 (x^2 +y^2) + x y \ge-\frac{25}{8}</math>. When x is set equal to y then there is no solution for x and y that satisfies the equality. However when taking <math>(x,y)=\left(\frac{\sqrt{5}}{2},-\frac{\sqrt{5}}{2} \right)</math> the equality does hold. This leads us to think that given a 2 var inequality for all reals, we find the equality case when the variables are additive inverses.
 
 
 
 
 
 
− ==Clarification of Domain of TPA's equality==
 
 
 
− *Consider <math>x^4+y^4 - 2 (x^2 +y^2) + x y+4 x^2 y^2 \ge-\frac{9}{8}</math>. When x is set equal to y then there is no solution for x and y that satisfies the equality. However when taking <math>(x,y)=\left( \frac{1}{4}\left(\sqrt{2}-\sqrt{6}\right), \frac{1}{4}\left(\sqrt{2}+\sqrt{6}\right) \right)</math> the equality does hold so it is a strict equality. This is because <math>- 2 (x^2 +y^2)</math> is not positive, and thus this does not count as an inequality involving positive parts.
 

Latest revision as of 20:48, 9 November 2018

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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