Difference between revisions of "2014 AMC 10A Problems/Problem 21"
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==Solution 2== | ==Solution 2== |
Revision as of 21:47, 11 June 2018
Contents
[hide]Problem
Positive integers and are such that the graphs of and intersect the -axis at the same point. What is the sum of all possible -coordinates of these points of intersection?
Solution 1
Note that when , the values of the equations should be equal by the problem statement. We have that Which means that The only possible pairs then are . These pairs give respective -values of which have a sum of .
It is curious when plugging each ordered pair into the equations, the pair of lines are the same. As an exercise, ask yourself why that is.
Solution 2
First, notice that the value of x cannot exceed 5 because the minimum value for a is 1. Also, notice that for the second equation, it intersects x at and so on. We then realize that the only integer values for x are and . We also see that for a fraction to be the value of x, the numerator must divide 5 evenly. So, the only other values are and . Adding, we get .
See Also
2014 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 20 |
Followed by Problem 22 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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