Greece National Olympiad Problem 4

Algebra Level 5

Let $a,b,c$ be positive real numbers with sum $6$ . Find the maximum value of $S = \sqrt[3]{{{a^2} + 2bc}} + \sqrt[3]{{{b^2} + 2ca}} + \sqrt[3]{{{c^2} + 2ab}}.$

This problem is part of this set .

The answer is 6.8682854553199912068482532.

1 solution

John Paul Babas
Mar 22, 2015

a=1 b=2 c=3 because the sum of a+b+c=6

then:

1st cube root
a^2=1
2bc=2(2)(3) = 12
1+12=13
cube root of 13 is 2.35

2nd cube root
b^2=4
2ca=2(3)(1)=6
4+ 6=10 cube root of 10 is 2.15

3rd cube root c^2=9 2ab=2(1)(2)=4 9+4=13 cube root of 13=2.35

then add: 2.35+2.15+2.35= 6.85 s=6.85

did i make it easy or complicated?

You can't just assume values for a, b, c.

Still, if you do want to assume values, you would have to take a=b=c

Vaibhav Prasad - 6 years, 2 months ago

Greece National Olympiad Problem 4

The answer is 6.8682854553199912068482532.

1 solution

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