# 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