An algebra problem by Quang Trần Minh

Algebra Level 3

Positive numbers a a , b b , and c c are such that a + b + c = 6 a+b+c=6 . Find the minimum value of

a + 1 + b + 1 + c + 1 a b + b c + c a + 15 . \frac{\sqrt{a+1} + \sqrt{b+1} + \sqrt{c+1}}{\sqrt{ab+bc+ca + 15}}.

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Quang Trần Minh by Quang Trần Minh , 19, Vietnam

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2 solutions

0 Helpful 1 Interesting 0 Brilliant 10 Confused

OK. But the solution needs to be written in English, not Vietnamese, man!

Anh Khoa Nguyễn Ngọc - 8 months, 2 weeks ago
Gaoyi Zhu
Oct 27, 2020

For the value to be the smallest, a, b, and c must all also be as small as possible. Hence, the smallest possible set of values of a, b, c which satisfies a+b+c=6 is: a=2; b=2; c=2 (Take note that the question did not mention that a, b, c cannot be of the same value!) Substituting these values into the expression given, we get 1 as the answer.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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