IMC 2016 Q19
19. A list of positive integers has a median of 8, a mode of 9 and a mean of 10.
What is the smallest possible number of integers in the list?
- A 5
- B 6
- C 7
- D 8
- E 9
Intermediate Mathematical Challenge 2016, Question 19. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B 6
From the information given, there are at least two 9s in the list, since 9 is the mode, and at least one number greater than 10, since 10 is the mean. So there are at least three numbers greater than 8 in the list. Therefore the list must contain at least six numbers, as the median of the numbers is 8. Moreover, it is possible to find suitable lists of six numbers with sum 60 (as the mean is 10), for example 1, 2, 7, 9, 9, 32.
Solution © UK Mathematics Trust, from the IMC 2016 solutions. Longer versions with investigations: Solutions and investigations (PDF).