2023 SSMO Team Round Problems/Problem 6

Problem

Suppose that $a,b,c$ are positive reals satisfying $(a^3+4)(b^3+6)(c^3+8) = 8(a+b+c)^3.$ Find the sum of all possible values of $\frac{bc}{a^2}.$ If you believe there are no solutions, put $0$ as your answer. If you believe the sum is infinity, put $1000$ as your answer.

Solution

From Hölder's Inequality, we have: $(a^3 + 2 + 2)(2 + b^3 + 4)(4 + 4 + c^3) \geq 8(a + b + c)^3.$

Equality holds when: $a^3 : 2 : 2 = 2 : b^3 : 4 = 4 : 4 : c^3.$

From the first and third proportions, we get $a^3 = 2$ and $c^3 = 2$ . From the first and second proportions, we get $a^3 = 1$ .

This is a contradiction, so equality cannot occur. Therefore, there are no real numbers $a, b, c$ satisfying the equality condition, and the answer is: $\boxed{0}.$

~SMO_Team

Page

Toolbox

Search

2023 SSMO Team Round Problems/Problem 6

Problem

Solution