Difference between revisions of "Mock AIME 3 Pre 2005 Problems/Problem 14"
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Since <math>PR</math> is the height of <math>\Delta O_1O_2O_3</math> to side <math>O_1O_2</math>, then using similar triangles, | Since <math>PR</math> is the height of <math>\Delta O_1O_2O_3</math> to side <math>O_1O_2</math>, then using similar triangles, | ||
− | <math>\frac{|O_1O_2|}{|PR|}=\frac{|AD|}{|h_1|}</math>. Therefore, <math>\frac{r_1+r_2}{16}=\frac{3}{\frac{2}{3}A_1}</math> | + | <math>\frac{|O_1O_2|}{|PR|}=\frac{|AD|}{|h_1|}</math>. Therefore, <math>\frac{r_1+r_2}{16}=\frac{3}{\frac{2}{3}A_1}</math>. Solving for <math>r_2</math> we have: |
<math>r_2=\frac{72}{A_1}-r_1=\frac{72}{A_1}-\frac{18}{A_1}=\frac{54}{A_1}=3\left( \frac{18}{A_1} \right)=3r_1</math> | <math>r_2=\frac{72}{A_1}-r_1=\frac{72}{A_1}-\frac{18}{A_1}=\frac{54}{A_1}=3\left( \frac{18}{A_1} \right)=3r_1</math> |
Revision as of 02:04, 25 November 2023
Problem
Circles and
are centered on opposite sides of line
, and are both tangent to
at
.
passes through
, intersecting
again at
. Let
and
be the intersections of
and
, and
and
respectively.
and
are extended past
and intersect
and
at
and
respectively. If
and
, then the area of triangle
can be expressed as
, where
and
are positive integers such that
and
are coprime and
is not divisible by the square of any prime. Determine
.
Solution
Let and
be the centers of
and
respectively.
Let point be the midpoint of
. Thus,
and
Let and
be the radii of circles
and
respectively.
Let and
be the areas of triangles
and
respectively.
Since and
, then
, and
This means that . In other words, those three triangles are similar.
Since is the circumcenter of
,
then
Let be the height of
to side
Then, , thus
Since is the height of
to side
, then using similar triangles,
. Therefore,
. Solving for
we have:
By similar triangles,
Using Heron's formula,
, where
we have:
, thus
~Tomas Diaz. orders@tomasdiaz.com
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See Also
Mock AIME 3 Pre 2005 (Problems, Source) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |