= Solution
The two incoming particles have equal energies and opposite <momenta>, so their total <four-momentum> in this frame is
$$
P_{\rm in}=\left(\frac{2E}{c},\mathbf0\right).
$$
It is already a <centre-of-momentum frame>. By <four-momentum conservation>, the final spatial momenta sum to zero and their energies sum to $2E$. Each product has energy
$$
E_i=\sqrt{\mu_i^2c^4+c^2|\mathbf p_i|^2}\geq\mu_ic^2.
$$
Summing gives the necessary <relativistic collision mass threshold>
$$
\boxed{E\geq\frac12\left(\sum_{i=1}^n\mu_i\right)c^2.}
$$
For massive products the equality threshold has all of them at rest in this frame. Above it, multiple products can carry balancing momenta and relative <kinetic energy>. The displayed condition is necessary, as requested; a single final particle must instead have exactly the incoming invariant mass, and additional physical conservation laws could further restrict a real reaction.
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