2018 AMC 12B Problem 10

Question

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?

202

223

224

225

234

The

2018-10=2008

non-mode entries can each appear at most

9

times

Find the fewest distinct non-mode values needed, then add

1

for the mode

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.

Problem 10 in other years