Difference between revisions of "2000 AIME I Problems/Problem 14"
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=== Solution 3 === | === Solution 3 === | ||
− | Let the measure of <math>\angle BAC</math> be <math>\alpha</math> and <math>\overline{AP}=\overline{PQ}=\overline{QB}=\overline{BC}=x</math>. Because <math>\triangle APQ</math> is isosceles, <math>AQ=2x\cos(\alpha | + | Let the measure of <math>\angle BAC</math> be <math>\alpha</math> and <math>\overline{AP}=\overline{PQ}=\overline{QB}=\overline{BC}=x</math>. Because <math>\triangle APQ</math> is isosceles, <math>AQ=2x\cos(\alpha)</math>. So, <math>\overline{AB}=x\left(2\cos(\alpha)+1\right)</math>. <math>\triangle{ABC}</math> is isosceles too, so <math>x=\overline{BC}=2\overline{AC}\sin\left(\frac{\alpha}{2}\right)=2x\left(2\cos(\alpha)+1\right)\sin\left(\frac{\alpha}{2}\right)</math>. |
Simplifying, <math>1=4\cos(\alpha)\sin\left(\frac{\alpha}{2}\right)+2sin\left(\frac{\alpha}{2}\right)</math>. By double angle formula, we know that <math>\cos(\alpha)=1-2\sin^2\left(\frac{\alpha}{2}\right)</math>. | Simplifying, <math>1=4\cos(\alpha)\sin\left(\frac{\alpha}{2}\right)+2sin\left(\frac{\alpha}{2}\right)</math>. By double angle formula, we know that <math>\cos(\alpha)=1-2\sin^2\left(\frac{\alpha}{2}\right)</math>. | ||
− | Applying, <math> | + | Applying, <math>4\sin\left(\frac{\alpha}{2}\right)-8\sin^3\left(\frac{\alpha}{2}\right)+2\sin\left(\frac{\alpha}{2}\right)=1</math> and <math>2\left(3\sin\left(\frac{\alpha}{2}\right)-4\sin^3\left(\frac{\alpha}{2}\right)\right)=1</math>. |
− | The expression in the parentheses though is triple angle formula! Hence, <math>sin\left(\frac{3\alpha}{2}\right)=\frac{1}{2}</math>, <math>\alpha=20^o</math>. It follows now that <math>\angle APQ=140^o</math>, <math>\angle ACB=80^o</math>. Giving <math>r=\frac{4}{7}. | + | The expression in the parentheses though is triple angle formula! Hence, <math>sin\left(\frac{3\alpha}{2}\right)=\frac{1}{2}</math>, <math>\alpha=20^o</math>. It follows now that <math>\angle APQ=140^o</math>, <math>\angle ACB=80^o</math>. Giving <math>r=\frac{4}{7}</math>. |
− | < | + | <math>\left\lfloor 1000\left(\frac {4}{7}\right)\right\rfloor = \boxed{571}</math>. |
== See also == | == See also == |
Revision as of 11:29, 26 December 2016
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
Solution 1
![[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); draw(A--B--C--A);draw(P--Q);draw(A--R--B);draw(P--R);D(R--C,dashed); 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)); [/asy]](http://latex.artofproblemsolving.com/f/1/f/f1f597c511a21d50292f1daf4ee93815c1404c82.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
![[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/1/6/6/166838a1bc564730cc23500919254edab0c132e2.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
.
.
Solution 3
Let the measure of be
and
. Because
is isosceles,
. So,
.
is isosceles too, so
.
Simplifying,
. By double angle formula, we know that
.
Applying,
and
.
The expression in the parentheses though is triple angle formula! Hence,
,
. It follows now that
,
. Giving
.
.
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 |
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