2000 AIME I Problems/Problem 14
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
.
Solution
![[asy]defaultpen(fontsize(8)); size(200); pair A=20*dir(80)+20*dir(60)+20*dir(100), B=(0,0), C=20*dir(0), P=20*dir(80)+20*dir(60), Q=20*dir(80), R=20*dir(60), S; S=intersectionpoint(Q--C,P--B); draw(A--B--C--A);draw(B--P--Q--C--R--Q);draw(A--R--B);draw(P--R--S); label("A",A,(0,1));label("B",B,(-1,-1));label("C",C,(1,-1));label("P",P,(1,1)); label("Q",Q,(-1,1));label("R",R,(1,0));label("S",S,(-1,0)); [/asy]](http://latex.artofproblemsolving.com/5/d/1/5d15b415bc4cc8bef1044cd299dd43fae4ea731b.png)
Let point be in
such that
, thus
is a rhombus and
is an isosceles trapezoid.
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 • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |