1979 USAMO Problems/Problem 4
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Problem
lies between the rays
and
. Find
on
and
on
collinear with
so that $\frac{1}{PQ}\plus{} \frac{1}{PR}$ (Error compiling LaTeX. Unknown error_msg) is as large as possible.[/
Solution
Perform the inversion with center and radius
Lines
go to the circles
passing through
and the line
cuts
again at the inverses
of
Hence
Thus, it suffices to find the line through that maximizes the length of the segment
If
are the midpoints of
i.e. the projections of
onto
then from the right trapezoid
we deduce that
Consequently,
is the greatest possible length of
which obviously occurs when
is a rectangle. Hence,
are the intersections of
with the perpendicular to
at