2022 AMC 12B Problem 7

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.