2018 AMC 12B Problem 10

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?

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Solution

The mode uses

10

of the entries, leaving

2008.

Because the mode is unique, every other value appears at most

9

times, so at least

\left\lceil\tfrac{2008}{9}\right\rceil=224

distinct non-mode values are needed. Adding the mode gives

224+1=225.

This is achievable: use

9

copies each of

1

through

223,

ten copies of

224,

and one copy of

225.

Thus, the correct answer is D.

Tagged: mode · pigeonhole principle