2020 CIME II Problems/Problem 8
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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:
![\[\frac{B}{A} = 16\frac{100-B}{100-A}\]](http://latex.artofproblemsolving.com/b/e/0/be085a12f58ce5754658ae1fcecf194727e5e8aa.png)
![\[(100-A)B = 16(100-B)A\]](http://latex.artofproblemsolving.com/3/1/4/3140c108239e38f9f075df9bd8898929e14e6f1c.png)
![\[100B - AB = 1600 A - 16AB\]](http://latex.artofproblemsolving.com/e/9/5/e95f24055adf9a3c8af0d02de311f9f818d4cdb0.png)
Let 
and 
:
![\[50C + 50D - \frac{D^2-C^2}{4} = 800D - 800C - 4(D^2-C^2)\]](http://latex.artofproblemsolving.com/b/1/4/b14149473a2ceafc3e5064d2e4f4ffa6d8bb239b.png)
![\[15D^2 -3000D = 15C^2-3400C\]](http://latex.artofproblemsolving.com/a/b/0/ab0292dfdf83a56db70bb3ecdd84c90e706a908f.png)
Complete the squares:
![\[15(D-100)^2 - 1500^2 = 15(C-1700/15)^2 - 1700^2/15\]](http://latex.artofproblemsolving.com/f/0/b/f0bdff5678a0d4e517482fc6192da5724ac4e0f9.png)
![\[(D-100)^2 + \frac{(17^2-15^2) \times 100^2}{15^2} = (C-1700/15)^2\]](http://latex.artofproblemsolving.com/0/6/0/06000ec6b73f65d385a1e3191bc9677cca1ce68e.png)
![\[C = \frac{1700}{15} \pm \sqrt{(D-100)^2 + \frac{8^2 \times 100^2}{15^2}}\]](http://latex.artofproblemsolving.com/d/a/b/daba2db1a5fab76dda9052f303ec3fd971b095af.png)
Note that 
so must use minus. This means that C is maximized if 
![\[C = \frac{1700}{15} - \frac{8 \times 100}{15} = 900/15=60\]](http://latex.artofproblemsolving.com/c/6/7/c675aa7cf6f5420946ea77dbfbd1cb18eb95db05.png)
is at a maximum 
