2014 AMC 10A Problems/Problem 21
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,
.
Video Solution by Richard Rusczyk
https://www.youtube.com/watch?v=-vZKwIazT08&list=PLyhPcpM8aMvKEM8u4Q-7Gi0rU5WU4WOb1&index=1
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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