Difference between revisions of "2003 AIME I Problems/Problem 7"
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u guys can edit the formatting if youd like to | u guys can edit the formatting if youd like to | ||
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== See also == | == See also == |
Revision as of 12:16, 21 March 2020
Problem
Point is on
with
and
Point
is not on
so that
and
and
are integers. Let
be the sum of all possible perimeters of
. Find
Solution
![[asy] size(220); pointpen = black; pathpen = black + linewidth(0.7); pair O=(0,0),A=(-15,0),B=(-6,0),C=(15,0),D=(0,8); D(D(MP("A",A))--D(MP("C",C))--D(MP("D",D,NE))--cycle); D(D(MP("B",B))--D); D((0,-4)--(0,12),linetype("4 4")+linewidth(0.7)); MP("6",B/2); MP("15",C/2); MP("9",(A+B)/2); [/asy]](http://latex.artofproblemsolving.com/1/5/b/15bb134453eeb3db27cdd27ad65a8176fe0393b3.png)
Denote the height of as
,
, and
. Using the Pythagorean theorem, we find that
and
. Thus,
. The LHS is difference of squares, so
. As both
are integers,
must be integral divisors of
.
The pairs of divisors of are
. This yields the four potential sets for
as
. The last is not a possibility since it simply degenerates into a line. The sum of the three possible perimeters of
is equal to
.
Solution 2
Using Stewart's Theorem, letting the side length be c, and the cevian be d, then we have
. Dividing both sides by thirty leaves
.
The solution follows as above.
Solution 3
drop an altitude from angle D to side AC. let the intersection point be E. since the triangle is isosceles, AE is half of AC, which is 15. Then, label side AD as x. since AED is a right triangle, you can figure out cos(A) with adjacent divided by hypotenuse, which in this case is AE divided by x, which is just 15/x. now law of cosines. label bd as y. so, by law of cosines,
Since cosine A is equal to 15/x,
, which can be simplified to
then continue with the first solution
u guys can edit the formatting if youd like to
-intelligence_20
See also
2003 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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