2000 AIME I Problems/Problem 14
From AoPSWiki
Problem
In triangle
it is given that angles
and
are congruent. Points
and
lie on
and
respectively, so that
Angle
is
times as large as angle
where
is a positive real number. Find the greatest integer that does not exceed
.
Contents |
Solution
Solution 1
![Click to view code [Asy_image]](http://alt1.artofproblemsolving.com/Forum/latexrender/pictures/6/2/a/62acd0bf993367d5ff70367359748e5d67af084a.png)
Let point
be in
such that
. Then
is a rhombus, so
and
is an isosceles trapezoid. Since
bisects
, it follows by symmetry in trapezoid
that
bisects
. Thus
lies on the perpendicular bisector of
, and
. Hence
is an equilateral triangle.
Now
, and the sum of the angles in
is
. Then
and
, so the answer is
.
Solution 2
![Click to view code [Asy_image]](http://alt2.artofproblemsolving.com/Forum/latexrender/pictures/c/d/a/cdaf0950c7719c8edd39d118683dfb8e3fb31445.png)
Again, construct
as above.
Let
and
, which means
.
is isosceles with
, so
.
Let
be the intersection of
and
. Since
,
is cyclic, which means
.
Since
is an isosceles trapezoid,
, but since
bisects
,
.
Therefore we have that
.
We solve the simultaneous equations
and
to get
and
.
,
, so
.
.
See also
| 2000 AIME I (Problems • Resources) | ||
| Preceded by Problem 13 | Followed by Problem 15 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||



