Problem 22 · 2012 AMC 8
Hard
Counting & Probability
median-window
Let R be a set of nine distinct integers. Six of the elements are 2, 3, 4, 6, 9, and 14. What is the number of possible values of the median of R?
Show answer
Answer: D — 7 possible values.
Show hints
Hint 1 of 2
With 9 distinct integers, the median is just the 5th smallest — the one with exactly 4 below it and 4 above. You get to add 3 free integers, so you can shove the "window" left or right. Ask: which values can't be reached no matter how you place those 3?
Still stuck? Show hint 2 →
Hint 2 of 2
Find the boundaries by counting what's already forced. A candidate median needs 4 below and 4 above; if too many of the six known values are already stuck on one side, you can't balance it. The technique is to find the largest and smallest reachable median, then count the integers between.
Show solution
Approach: find the median's reachable range, then count integers in it
- Median = 5th smallest of the 9. To make a value m the median, the final set must have 4 elements below m and 4 above. You control only 3 new integers, so check whether the six knowns (2, 3, 4, 6, 9, 14) can be balanced around m.
- Too small fails: if m < 3, then 3, 4, 6, 9, 14 — five knowns — already sit above m, more than the 4 allowed. So m ≥ 3.
- Too big fails: if m > 9, then 2, 3, 4, 6, 9 — five knowns — already sit below, again too many. So m ≤ 9.
- Every integer from 3 to 9 does work: place enough of your 3 spare integers below or above to even the count to 4 and 4. That's the integers 3, 4, 5, 6, 7, 8, 9 — 9 − 3 + 1 = 7 possible medians.
- The reusable idea: when you control some of the data, find the extreme positions the answer can reach (the boundaries) and count between them — don't test every value blindly.
Mark:
· log in to save