System of Three Equations

Algebra Level 3

{ a 2 + a b + b 2 3 = 2 b 2 3 + c 2 = 1 c 2 + c a + a 2 = 1 \begin{cases} a^2+ab+\dfrac{b^2}{3}=2\\ \dfrac{b^2}{3}+c^2=1\\ c^2+ca+a^2=1 \end{cases}

Given that positive real numbers a a , b b , and c c satisfy the system of equations above and a b + 2 b c + 3 c a = m n ab + 2bc + 3ca = m\sqrt{n} , where m m and n n are positive integers with n n being square-free, what is m + n m + n ?


The answer is 5.

Sophie Ho by Sophie Ho , 53, USA

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1 solution

Solving the given equations, we get a^2=(28-8√3)/37, b^2=(48+18√3)/37 and c^2=(21-6√3)/37. From this, we get ab=(30+2√3)/37, bc=(3+15√3)/37, and ca=(14√3-12)/37. Hence ab+2bc+3ca=2√3.

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