Skip to main content

2018 AMC 10B Problem 14

Problem 14 of 25IntermediateAlgebra

A list of 20182018 positive integers has a unique mode, which occurs exactly 1010 times. What is the least number of distinct values that can occur in the list?

Answer choices

Show solution

Solution

The mode shows up 1010 times. To keep the number of distinct values small, let every other value repeat as much as the rules allow, which is 99 times each (any more would tie the mode). With dd distinct values the list holds at most 10+9(d1)10 + 9(d-1) entries. We need 10+9(d1)2018,10 + 9(d-1) \ge 2018, so d1223.1,d - 1 \ge 223.1, giving d225.d \ge 225. Therefore, the answer is D.

More practice

Concepts: mode · optimization

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.