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Problem 230

Combinatorics Difficulty 1.7 Multiple choice CEMC Gauss (Grade 8) · Canada · 2012

At the Gaussland Olympics there are 480 student participants.
Each student is participating in 4 different events.
Each event has 20 students participating and is supervised by 1 adult coach.
There are 16 adult coaches and each coach supervises the same number of events.
How many events does each coach supervise?

Pick one

Official solution

In an ordered list of five integers, the median is the number in the middle or third position.

Thus if we let the set of integers be $a,b,c,d,e$ , ordered from smallest to largest, then $c=18$ .

Since the average is fixed (at 20), $e$ (the largest number in the set) is largest when $a,b$ and $d$ are as small as possible.

Since the numbers in the set are different positive integers, the smallest that $a$ can be is 1 and the smallest that $b$ can be is 2.

Our set of integers is now $1,2,18,d,e$ .

Again, we want $d$ to be as small as possible, but it must be larger than the median 18.

Therefore, $d=19$ .

Since the average of the 5 integers is 20, then the sum of the five integers is $5\times20=100$ .

Thus, $1+2+18+19+e=100$ or $40+e=100$ , and so $e=60$ .
The greatest possible integer in the set is 60.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.