# Difference between revisions of "TPA's equality"

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− | '''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 <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> | |

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− | + | ==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. | |

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− | + | ==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]]) | |

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− | + | [[Category:Inequality]] | |

− | + | [[Category:Theorems]] | |

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## 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 be any symmetric inequality. Let be the parts of that inequality. Then in contest space, achieves equality when

## 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 , we have the maximum of is .

### Olympiad

- Let be positive real numbers. Prove that the minimum of is (Source)