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Difference between revisions of "Mock AIME 3 Pre 2005 Problems/Problem 14"
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Using Heron's formula,
 
Using Heron's formula,
  
−
<math>A_1=sqrt{p(p-a)(p-b)(p-c)}</math>, where <math>p=\frac{a+b+c}{2}=\frac{3+6+4}{2}=\frac{13}{2}</math> we have:
+
<math>A_1=sqrt{s(s-a)(s-b)(s-c)}</math>, where <math>s=\frac{a+b+c}{2}=\frac{3+6+4}{2}=\frac{13}{2}</math> we have:
  
 
<math>A_1=sqrt{\left( \frac{13}{2} \right)\left( \frac{13}{2}-3\right)\left( \frac{13}{2}-6 \right)\left( \frac{13}{2}-4 \right)}</math>
 
<math>A_1=sqrt{\left( \frac{13}{2} \right)\left( \frac{13}{2}-3\right)\left( \frac{13}{2}-6 \right)\left( \frac{13}{2}-4 \right)}</math>

Revision as of 00:42, 25 November 2023

Problem

Circles $\omega_1$ and $\omega_2$ are centered on opposite sides of line $l$, and are both tangent to $l$ at $P$. $\omega_3$ passes through $P$, intersecting $l$ again at $Q$. Let $A$ and $B$ be the intersections of $\omega_1$ and $\omega_3$, and $\omega_2$ and $\omega_3$ respectively. $AP$ and $BP$ are extended past $P$ and intersect $\omega_2$ and $\omega_1$ at $C$ and $D$ respectively. If $AD = 3, AP = 6, DP = 4,$ and $PQ = 32$, then the area of triangle $PBC$ can be expressed as $\frac{p\sqrt{q}}{r}$, where $p, q,$ and $r$ are positive integers such that $p$ and $r$ are coprime and $q$ is not divisible by the square of any prime. Determine $p + q + r$.

Solution

Let $O_1, O_2,$ and $O_3$ be the centers of $\omega_1, \omega_2$ and $\omega_3$ respectively.

Let point $R$ be the midpoint of $QP$. Thus, $O_3R \bot PQ$ and $|PR|=\frac{|PR|}{2}=16$

Let $r_1$ and $r_2$ be the radii of circles $\omega_1$ and $\omega_2$ respectively.

Let $A_1$ and $A_2$ be the areas of triangles $PDA$ and $PBC$ respectively.

Since $O_3R \parallel O_1O_2$ and $\angle RO_3O_2 = \angle O_3O_2O_1$, then $O_2B \parallel O_1D$, and $BC \parallel AD$

This means that $\Delta PDA \sim \Delta PBC \sim \Delta PDA O_3O_1O_1$. In other words, those three triangles are similar.

Since $r_1$ is the circumcenter of $\Delta PDA$,

then $r_1=\frac{|AD| \times |DP| \times |AP|}{4A_1}=\frac{(3)(4)(6)}{4A_1}=\frac{18}{A_1}$

Let $h_1$ be the height of $\Delta PDA$ to side $AD$

Then, $A_1=\frac{|AD| \times h_1}{2}$, thus $h_1=\frac{2}{3}A_1$

Using similar triangles, $\frac{|O_1O_2|}{|PR|}=\frac{|AD|}{|h_1|}$

Therefore, $\frac{r_1+r_2}{16}=\frac{3}{\frac{2}{3}A_1}$

$r_2=\frac{72}{A_1}-r_1=\frac{72}{A_1}-\frac{18}{A_1}=\frac{54}{A_1}=3\left( \frac{18}{A_1} \right)=3r_1$

By similar triangles,

$A_2=\left( \frac{r_2}{r_1} \right)^2A_1=9A_1$

Using Heron's formula,

$A_1=sqrt{s(s-a)(s-b)(s-c)}$, where $s=\frac{a+b+c}{2}=\frac{3+6+4}{2}=\frac{13}{2}$ we have:

$A_1=sqrt{\left( \frac{13}{2} \right)\left( \frac{13}{2}-3\right)\left( \frac{13}{2}-6 \right)\left( \frac{13}{2}-4 \right)}$



Invert about a circle with radius 1 and center P. Note that since all relevant circles and lines go through P, they all are transformed into lines, and $\omega_1,\omega_2, l$ are all tangent at infinity (i.e. parallel). That was the crux move; some more basic length chasing using similar triangles gets you the answer.

~Tomas Diaz. orders@tomasdiaz.com

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
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