A.M-G.M Inequality

Algebra Level 4

Let a , b a,b and c c be positive numbers satisfying a + b + c = 3 a+b+c=3 . If the maximum value of a 2 b 3 c 2 a^2 b^3 c^2 can be expressed as 3 p × 2 q × 7 r 3^p \times 2^q \times 7^r , where p , q , r p,q,r are integers, find the value of p + q + r p+q+r .


The answer is 7.

Abhiram Rao by Abhiram Rao , 18, India

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1 solution

Md Zuhair
Dec 7, 2016

By AM-GM, 3 = a + b + c = a 2 + a 2 + b 3 + b 3 + b 3 + c 2 + c 2 ≥ 7 a 2 b 3 c 2 2 4 ⋅ 3 3 7 , 3 = a + b + c = \frac{a}{2} + \frac{a}{2} + \frac{b}{3} + \frac{b}{3} + \frac{b}{3} + \frac{c}{2} + \frac{c}{2} \ge 7 \sqrt[7]{\frac{a^2 b^3 c^2}{2^4 \cdot 3^3}}, so a 2 b 3 c 2 ≤ 2 4 ⋅ 3 10 ⋅ 7 − 7 . a^2 b^3 c^2 \le 2^4 \cdot 3^{10} \cdot 7^{-7}.

Equality occurs when a = 6 / 7 a = 6/7 , b = 9 / 7 b = 9/7 , c = 6 / 7 c = 6/7 , so p + q + r = 4 + 10 + ( − 7 ) = 7 p + q + r = 4 + 10 + (-7) = 7 .

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