03/17/2021 (2011: Countdown Round, Problem 25)

Q: A list of five positive integers has all of the following properties:

• the only integer that occurs more than once is 80,
• the median is 90 and
• the mean is 100.

What is the largest possible integer that could appear in this list?

A: 159

Using the first two bullets of information we have, we can write out the numbers we have, and we get [80, 80, 90, x, y], where x and y are unknown integers. 

Next, we use the third bullet. The mean is just the sum of the numbers divided by the total amount of numbers. That means that the sum of the list is 5 ⋅ 100 = 500.

500 = 80 + 80 + 90 + x + y

We can subtract (80 + 80 + 90) from both sides of the above equation to get: 250 = x + y.

If we are trying to find the greatest number that can be on this list, we have to use the smallest value of x (to maximize the value of y). Because 90 isn't repeated (only 80 is repeated), and all the numbers are integers, that means 91 is the smallest x-value, and 250 - 91 = 159 is the greatest y-value. 

That means that 159 is our answer