2018 AMC 12B Problem #10
Probability
Level
2
A list of 2018 positive integers has a unique mode, which occurs exactly 10 times. What is the least number of distinct values that can occur in the list?
This problem is part of the 2018 AMC 12B .
234
224
223
202
225
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1 solution
The mode appears 10 times, so there are 2 0 1 8 − 1 0 = 2 0 0 8 numbers left to distribute among other values.
To get the smallest number of distinct values, we need to have each appear as many times as possible, and that is 9 times at most.
9 2 0 0 8 = 2 2 3 + 9 1
This can be covered by 2 2 3 + 1 = 2 2 4 values.
Adding the mode makes a total of 2 2 5 values.