Problem:
The table below displays some of the results of last summer's Frostbite Falls Fishing Festival, showing how many contestants caught n fish for various values of n.
n number of contestants who caught n fish 091527323⋯⋯135142151
In the newspaper story covering the event, it was reported that
a) the winner caught 15 fish;
b) those who caught 3 or more fish averaged 6 fish each;
c) those who caught 12 or fewer fish averaged 5 fish each.
What was the total number of fish caught during the festival?
Solution:
Let F be the total number of fish caught during the festival and C be the total number of contestants. Then C−(9+5+7)=C−21 contestants each caught 3 or more fish, and these contestants caught a total of F−(0⋅9+1⋅5+2⋅7)=F−19 fish. Hence
C−21F−19=6(1)
Similarly, C−(5+2+1)=C−8 contestants each caught 12 or fewer fish, and these contestants caught a total of F−(5⋅13+2⋅14+1⋅15)=F−108 fish. Thus
C−8F−108=5(2)
Solving (1) and (2) simultaneously, we find C=175 and F=943.
The problems on this page are the property of the MAA's American Mathematics Competitions