2016 TIMC keystage3-IWYMIC individual Question 13

Let $a$, $b$ and $c$ be positive real numbers such that $\frac{8a^2}{a^2+9}=b$, $\frac{10b^2}{b^2+16}=c$ and $\frac{6c^2}{c^2+25}=a$. Find $a+b+c$.

Submission guidance: Include a detailed explanation, as required for Section B.