2018 AMC 10B Problems/Problem 14

The following problem is from both the 2018 AMC 12B #10 and 2018 AMC 10B #14, so both problems redirect to this page.

Problem

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?

$\textbf{(A)}\ 202\qquad\textbf{(B)}\ 223\qquad\textbf{(C)}\ 224\qquad\textbf{(D)}\ 225\qquad\textbf{(E)}\ 234$

Solution

To minimize the number of distinct values, we want to maximize the number of times they appear. So, we could have $223$ numbers appear $9$ times, $1$ number appear once, and the mode appear $10$ times, giving us a total of $223 + 1 + 1$ = $\boxed{\textbf{(D) } 225}$

Solution 2 (Setting up an Equation)

As in the previous solution, we want to maximize the number of time each number appears to do so, we can set up an equation $10 + 9( x - 1 )$ ≥ $2018$ where $x$ is the number of values. Notice how we can then rearrange the equation into $1 + 9 ( 1 )+9 ( x - 1 )$ ≥ $2018$ which becomes $9 x$ ≥ $2017$ then $x$ ≥ $224$ $1/9$ . We cannot have a fraction of a value so we must round up to $225$ . The solution is $\boxed{\textbf{(D) } 225}$

Solution by Username_taken12

2018 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
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All AMC 10 Problems and Solutions

2018 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 9	Followed by Problem 11
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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2018 AMC 10B Problems/Problem 14

Contents

Problem

Solution

Solution 2 (Setting up an Equation)

See Also