Olympiad problems

2018 AMC 12/AHSME, 10

Tags: mode

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

2000 AMC 12/AHSME, 14

Tags: arithmetic sequence , list , mean , median , mode

When the mean, median, and mode of the list \[ 10, 2, 5, 2, 4, 2, x\]are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of $ x$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 17 \qquad \textbf{(E)}\ 20$

2016 AMC 10, 7

Tags: arithmetic mean , mode , median

The mean, median, and mode of the $7$ data values $60, 100, x, 40, 50, 200, 90$ are all equal to $x$. What is the value of $x$? $\textbf{(A)}\ 50 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 75 \qquad\textbf{(D)}\ 90 \qquad\textbf{(E)}\ 100$

2010 AMC 8, 4

Tags: statistics , mean , median , mode , arithmetic mean

What is the sum of the mean, median, and mode of the numbers, $2,3,0,3,1,4,0,3$? $ \textbf{(A)}\ 6.5 \qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 7.5\qquad\textbf{(D)}\ 8.5\qquad\textbf{(E)}\ 9 $

2010 Contests, 4

Tags: statistics , mean , median , mode , arithmetic mean

What is the sum of the mean, median, and mode of the numbers, $2,3,0,3,1,4,0,3$? $ \textbf{(A)}\ 6.5 \qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 7.5\qquad\textbf{(D)}\ 8.5\qquad\textbf{(E)}\ 9 $

2018 AMC 10, 14

Tags: mode

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