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# 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.

Note by Siddharth G
2 years, 11 months ago

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

- 2 years, 11 months ago

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

- 2 years, 11 months ago