NMTC 2nd Level 2014

Let ${ a }_{ 1 }{ a }_{ 2 }{ ...a }_{ 2014 }$ be a random arrangement of $1,2,3...2014$. Prove that

$\frac { 1 }{ { a }_{ 1 }{ +a }_{ 2 } } +\frac { 1 }{ { a }_{ 2 }+{ a }_{ 3 } } +.....\frac { 1 }{ { a }_{ 2012 }{ +a }_{ 2013 } } +\frac { 1 }{ { a }_{ 2013 }{ +a }_{ 2014 } } >\frac { 2013 }{ 2016 }$

This a part of my set NMTC 2nd Level (Junior) held in 2014.

#Algebra #Inequalities #NMTC

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Comments

Hint: Apply Titu-lemma(Cauchy inequality) , and then use $a_1+a_{2014} \ge 3$

Shivang Jindal - 6 years, 7 months ago

Thanks! I also found an AM-HM solution along the way.

Siddharth G - 6 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$