2021 AMC 10B Problems/Problem 19
Problem
Suppose that is a finite set of positive integers. If the greatest integer in
is removed from
, then the average value (arithmetic mean) of the integers remaining is
. If the least integer is
is [i]also[/i] removed, then the average value of the integers remaining is
. If the greatest integer is then returned to the set, the average value of the integers rises of
. The greatest integer in the original set
is
greater than the least integer in
. What is the average value of all the integers in the set
Solution
Let the lowest value be L and the highest G, and let the sum be Z and the amount of numbers n. We have ,
,
, and
. Clearing denominators gives
,
, and
. We use
to turn the first equation into
, which gives
. Turning the second into
we see
and
so the average is
~aop2014