1993 AIME Problems/Problem 3
The table below displays some of the results of last summer's Frostbite Falls Fishing Festival, showing how many contestants caught fish for various values of .
In the newspaper story covering the event, it was reported that
- (a) the winner caught fish;
- (b) those who caught or more fish averaged fish each;
- (c) those who caught or fewer fish averaged fish each.
What was the total number of fish caught during the festival?
Suppose that the number of fish is and the number of contestants is . The fishers that caught or more fish caught a total of fish. Since they averaged fish,
Similarily, those who caught or fewer fish averaged fish per person, so
Solving the two equation system, we find that and , the answer.
Let be the total number of fish caught by the contestants who didn't catch , or fish and let be the number of contestants who didn't catch , or fish. From , we know that . From we have . Using these two equations gets us . Plug this back into the equation to get . Thus, the total number of fish caught is - Heavytoothpaste
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