2018 AMC 12A Problem 6

Problem 6 of 25EasierAlgebra

For positive integers

m

and

n

such that

m + 10 \lt n + 1,

both the mean and the median of the set

\{m, m + 4, m + 10,

n + 1,

n + 2,

2n\}

are equal to

n.

What is

m + n?

Show solution

Solution

Because

m + 10 \lt n + 1,

the six numbers are already increasing, so the median is the average of the middle two:

\frac{(m+10)+(n+1)}{2} = n,

giving

m = n - 11.

The mean condition is

\tiny \frac{(n-11)+(n-7)+(n-1)+(n+1)+(n+2)+2n}{6} = n,

7n - 16 = 6n

and

n = 16.

Then

m = 5,

and

m + n = 21.

Thus, the correct answer is B.