2020 CIME II Problems/Problem 8
Problem 8
A committee has an oligarchy, consisting of of the members of the committee. Suppose that
of the work is done by the oligarchy. If the average amount of work done by a member of the oligarchy is
times the amount of work done by a nonmember of the oligarchy, find the maximum possible value of
.
Solution
Average work done sets up an equation:
Let
and
:
Complete the squares:
Note that so must use minus. This means that C is maximized if
is at a maximum
Solution 2
As in the first solution, we get . We rearrange and obtain
. We divide by
to obtain
. We then subtract
from both sides, and factor to obtain
. If we graph this with
being on the
-axis and
being on the
-axis, this equation is the hyperbola
, except scaled up by
and translated
to the left and
up. This graph intersects
and
, and the maximum difference clearly occurs at the point when the slope of the function is
. This is at
. Our answer is
.
~mathboy100