= Solution
For a <proper lifetime inferred from a delayed decay flash>, the measured interval ends when the light returns to the detector, not when the <muon> decays. Put passage at $(t,x)=(0,0)$ in the detector frame. If decay occurs at $t_d$, its distance is $vt_d$, and the backwards <photon> takes another $vt_d/c$ to arrive. Therefore
$$
t_{\rm record}=t_d(1+v/c),\qquad t_d=\frac{10\,\mathrm{ns}}{1+1/2}=\frac{20}{3}\,\mathrm{ns}.
$$
The <muon>'s proper elapsed time is
$$
\boxed{\Delta\tau=t_d\sqrt{1-v^2/c^2}
=10\,\mathrm{ns}\sqrt{\frac{1-v/c}{1+v/c}}
=\frac{10}{\sqrt3}\,\mathrm{ns}\approx5.77\,\mathrm{ns}.}
$$
Applying <time dilation> directly to the 10 ns reception interval would transform the wrong pair of events.
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