2014 AIME I Problems/Problem 6
Problem 6
The graphs and
have y-intercepts of
and
, respectively, and each graph has two positive integer x-intercepts. Find
.
Solution
Begin by setting to 0, then set both equations to
and
, respectively. You'll notice that because the two parabolas have to have x-intercepts,
.
You'll know that , so you now need to find a positive integer
which has [b]positive integer[/b] x-intercepts for both equations.
Notice that if is -2 times a square number, then you have found a value of
for which the second equation has positive x-intercepts. We guess and check
to obtain
.
Following this, we check to make sure the first equation also has positive x-intercepts (which it does), so we can conclude the answer is .
See also
2014 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.