2022 AMC 12A Problem 6

Problem 6 of 25EasierAlgebra

A data set consists of

6

(not distinct) positive integers:

1,

7,

5,

2,

5,

and

X.

The average (arithmetic mean) of the

6

numbers equals a value in the data set. What is the sum of all positive values of

X?

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Solution

The known numbers sum to

20,

so the mean is

\dfrac{20+X}{6},

which must equal an element of the set. Setting it to

5

gives

X=10;

7

gives

X=22;

and to

X

itself gives

20+X=6X,

X=4.

Values

1

and

2

give negative

X.

The positive values are

10,22,4,

summing to

36.

Thus, the correct answer is D.