2022 AMC 12B Problem 7

Question

Camila writes down five positive integers. The unique mode of these integers is

2

greater than their median, and the median is

2

greater than their arithmetic mean. What is the least possible value for the mode?

5

7

9

11

13

The mode exceeds the median, so it can only appear among the two largest numbers

Let the median be

m;

for a unique mode the two largest must both equal

m+2,

and the sum is

5(m-2)

List the numbers in increasing order with median

m.

The mode is

m + 2 \gt m,

so it can only occur among the two largest entries; for it to be the unique mode, both of them must equal

m + 2.

The mean is

m - 2,

so the total is

5(m-2).

With the two largest equal to

m + 2

and the median

m,

the two smallest sum to

5(m-2) - m - 2(m+2)

= 2m - 14.

The two smallest are distinct positive integers, so

2m - 14 \ge 1 + 2 = 3,

giving

m \ge 9.

With

m = 9

the list

1, 3, 9, 11, 11

works, so the least mode is

m + 2 = 11.

Thus, the correct answer is D.

Problem 7 in other years