2000 AMC 12 Problem 14

Question

When the mean, median, and mode of the list

10, 2, 5, 2, 4, 2, x

are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of

x?

3

6

9

17

20

The mode is

2,

and the mean is

\dfrac{25 + x}{7}

Split into cases by where

x

falls relative to the other numbers, and require a non-constant progression

The six fixed numbers sum to

25,

so the mean is

\dfrac{25 + x}{7},

and the mode is

2.

If

x \le 2,

then

2

is both median and mode, forcing a constant progression, so

x \gt 2.

Case $2 \lt x \lt 4$ : the median is

x.

Requiring

2, x, \dfrac{25 + x}{7}

to form an arithmetic progression yields

x = 3

as the only value in this range.

Case $x \ge 4$ : the median is

4,

and the progression

2, 4, 6

forces the mean to be

6,

so

\dfrac{25 + x}{7} = 6,

giving

x = 17.

The sum of all possible values is

3 + 17 = 20.

Thus, the correct answer is E.

Problem 14 in other years