Difference between revisions of "2014 AIME I Problems/Problem 6"
(→Solution) |
(→Solution) |
||
Line 4: | Line 4: | ||
== Solution == | == Solution == | ||
− | Begin by setting <math>x</math> to 0, then set both equations to <math>h^2=\frac{2013-j}{3}</math> and <math>h^2=\frac{2014-k}{2}</math>, respectively. You'll notice that because the two parabolas have to have x-intercepts, <math>h\ | + | Begin by setting <math>x</math> to 0, then set both equations to <math>h^2=\frac{2013-j}{3}</math> and <math>h^2=\frac{2014-k}{2}</math>, respectively. You'll notice that because the two parabolas have to have x-intercepts, <math>h\ge32</math>. |
You'll know that <math>h^2=\frac{2014-k}{2}</math>, so you now need to find a positive integer <math>h</math> which has positive integer x-intercepts for both equations. | You'll know that <math>h^2=\frac{2014-k}{2}</math>, so you now need to find a positive integer <math>h</math> which has positive integer x-intercepts for both equations. |
Revision as of 13:48, 15 March 2014
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 positive integer 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.