Minimize the Expression 2

Algebra Level 4

Given that a , b , c > 0 a, b, c > 0 , minimize: a 2 + 1 2 a + b + b 2 + 1 2 b + c + c 2 + 1 2 c + a . \dfrac{a^2+1}{2a+b}+ \dfrac{b^2+1}{2b+c}+ \dfrac{c^2+1}{2c+a}.


The answer is 2.

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Alan Yan by Alan Yan , 20, USA

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1 solution

Alan Yan
Sep 8, 2015

Break apart the expression like so:

∑ c y c 1 2 a + b + ∑ c y c a 2 2 a + b ≥ 3 a + b + c + a + b + c 3 ≥ 2 (By T2/Cauchy and AM-GM) \sum_{cyc}{\frac{1}{2a+b}} + \sum_{cyc}{\frac{a^2}{2a+b}} \geq \frac{3}{a+b+c}+\frac{a+b+c}{3} \geq 2 \text{ (By T2/Cauchy and AM-GM)}

Equality holds when a = b = c = 1 a = b = c = 1 .

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Did the exact same. Nice question

Aditya Kumar - 5 years ago

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