Inequality (5)

Algebra Level 3

1 1 a + 1 + 1 b + 1 + 1 c + 1 − 1 1 a + 1 b + 1 c ≥ m n \large \frac{1}{\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}} - \frac{1}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}} \ge \frac{m}{n}

The inequality above holds true for positive reals a a , b b , and c c , and coprime positive integers m m and n n .

What is m + n = ? m+n=?

3 5 6 4 7

Hana Wehbi by Hana Wehbi , 51, USA

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2 solutions

Hana Wehbi
Mar 6, 2018

Expanding the inequality yields:

1 a ( a + 1 ) + 1 b ( b + 1 ) + 1 c ( c + 1 ) ≥ 1 3 ( 1 a + 1 b + 1 c ) ( 1 a + 1 + 1 b + 1 + 1 c + 1 ) ≥ 1 3 \frac{1}{a(a+1)}+\frac{1}{b(b+1)}+\frac{1}{c(c+1)} \ge \frac{1}{3}(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})(\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1})\ge \frac{1}{3}

This is true by Chebyshev's inequality, ⟹ m = 1 and n = 3 ⟹ m + n = 4 \implies m=1 \text{ and } n=3\implies m+n=4

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Lucas Machado
Mar 5, 2018

Just use AM>HM

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