Difference between revisions of "1995 AIME Problems/Problem 9"
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Revision as of 18:30, 4 July 2013
Contents
Problem
Triangle 
is isosceles, with 
and altitude 
Suppose that there is a point 
on 
with 
and 
Then the perimeter of 
may be written in the form 
where 
and 
are integers. Find 
![[asy] import graph; size(5cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-1.55,xmax=7.95,ymin=-4.41,ymax=5.3; draw((1,3)--(0,0)); draw((0,0)--(2,0)); draw((2,0)--(1,3)); draw((1,3)--(1,0)); draw((1,0.7)--(0,0)); draw((1,0.7)--(2,0)); label("$11$",(1,1.63),W); dot((1,3),ds); label("$A$",(1,3),N); dot((0,0),ds); label("$B$",(0,0),SW); dot((2,0),ds); label("$C$",(2,0),SE); dot((1,0),ds); label("$M$",(1,0),S); dot((1,0.7),ds); label("$D$",(1,0.7),NE); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);[/asy]](http://latex.artofproblemsolving.com/5/f/6/5f6666ff2f58f480f55c1a1ff6cd1cc54280dfcd.png)
Solution 1
Let 
, so 
. Then, 
. Expanding 
using the angle sum identity gives ![\[\tan 3x=\tan(2x+x)=\frac{3\tan x-\tan^3x}{1-3\tan^2x}.\]](http://latex.artofproblemsolving.com/6/1/d/61dd7327d2ec20f5a8c75e6ed0a3b2892b99142a.png)
Thus, 
. Solving, we get 
. Hence, 
and 
by the Pythagorean Theorem. The total perimeter is 
. The answer is thus 
.
Solution 2
In a similar fashion, we encode the angles as complex numbers, so if 
, then 
and 
. So we need only find 
such that 
. This will happen when 
, which simplifies to 
. Therefore, 
. By the Pythagorean Theorem, 
, so the perimeter is 
, giving us our answer, 
.
See also
| 1995 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 8 |
Followed by Problem 10 | |
| 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. 
