The arithmetic mean of a list of numbers is the sum of them divided by the number of elements in the list. For example, the arithmetic mean of the list 3, 3, 4, 5, and 10 is

$$
\frac{3+3+4+5+10}{5}=5
$$

The arithmetic mean of 5 distinct positive integers is equal to 11. What is the maximum possible value of a number in this list?

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Solution

Since the average of the 5 integers is 11, their sum is $5 \cdot 11=55$. As all are distinct positive integers, the sum of four of them is at least $1+2+3+4=10$. Therefore, the fifth element is at most $55-10=45$. Thus, the largest possible value of a number in this list is 45, and an example where this occurs is with the list 1, 2, 3, 4, 45.

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