Problem source

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\noindent In the diagram $P_{1}$ and $P_{2}$ are smooth light pulleys
fixed at the same height, and $P_{3}$ is a third smooth light pulley,
freely suspended. A smooth light inextensible string runs over $P_{1},$
under $P_{3}$ and over $P_{2},$ as shown: the parts of the string
not in contact with any pulley are vertical. A particle of mass $m_{3}$
is attached to $P_{3}.$ There is a particle of mass $m_{1}$ attached
to the end of the string below $P_{1}$ and a particle of mass $m_{2}$
attached to the other end, below $P_{2}.$ The system is released
from rest. Find the tension in the string, and show that the pulley
$P_{3}$ will remain at rest if 
\[
4m_{1}m_{2}=m_{3}(m_{1}+m_{2}).
\]