# 2020 CIME II Problems/Problem 8

## Contents

## 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

## Solution 3

Assume WLOG, there are members and that there are work that is done. So, there are members of the oligarchy and they've done work in total. Thus, on average, each member of the oligarchy does work. Then, the average work a nonmember of the oligarchy will be . Thus, When and , we have , giving

~sml1809