Difference between revisions of "2014 AMC 10A Problems/Problem 21"
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Revision as of 17:55, 21 October 2021
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
![\[0 = ax + 5 \implies x = -\dfrac{5}{a}\]](http://latex.artofproblemsolving.com/9/b/5/9b5974079f962de0b0cdffd7cc3e068e6aa9753a.png)
![\[0 = 3x+b \implies x= -\dfrac{b}{3}\]](http://latex.artofproblemsolving.com/0/5/d/05ddc94c34a7ca9558c5927b090cbe03dbd143a6.png)
Which means that ![\[-\dfrac{5}{a} = -\dfrac{b}{3} \implies ab = 15\]](http://latex.artofproblemsolving.com/9/a/8/9a816e2e78ddb95d2446fe65f792d4d65d844642.png)
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, .
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
