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.

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TopNewestHint:Apply Titu-lemma(Cauchy inequality) , and then use \( a_1+a_{2014} \ge 3 \)Log in to reply

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

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