2014 AMC 10A Problems/Problem 21
Contents
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 .
Solution 2
Going off of Solution 1, for the first equation, notice that the value of cannot be less than . We also know for the first equation that the values of have to be divided by something. Also, for the second equation, the values of can only be . Therefore, we see that, the only values common between the two sequences are , and adding them up, we get for our answer, .
Solution 3
Similar to the above solutions except we're using the equations and With this, we know that c has to be negative. Doing some math, we find to be and
Adding them up gives you our answer:.
~Starshooter11
Video Solution by Richard Rusczyk
https://artofproblemsolving.com/videos/amc/2014amc10a/375
~ dolphin7
See Also
2014 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 20 |
Followed by Problem 22 | |
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