1998 AIME Problems/Problem 12

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Problem

Let $ABC$ be equilateral, and $D, E,$ and $F$ be the midpoints of $\overline{BC}, \overline{CA},$ and $\overline{AB},$ respectively. There exist points $P, Q,$ and $R$ on $\displaystyle \overline{DE}, \overline{EF},$ and $\overline{FD}, \displaystyle$ respectively, with the property that $P$ is on $\overline{CQ}, Q$ is on $\overline{AR}, \displaystyle$ and $R$ is on $\overline{BP}.$ The ratio of the area of triangle $ABC$ to the area of triangle $PQR$ is $a + b\sqrt {c}, \displaystyle$ where $a, b$ and $c$ are integers, and $c$ is not divisible by the square of any prime. What is $a^{2} + b^{2} + c^{2}$?

1998 AIME-12.png

Solution

Assign variables: $EP = FQ = x$ $EQ = y$ $PQ = k$ Since $\displaystyle AE = \frac {1}{2}AB$ and $AD = \frac {1}{2}AC$, $\triangle AED \sim \triangle ABC$ and $ED \parallel BC$

Alternate Interior Angles Theorem says $\displaystyle \angle PEQ = \angle BFQ$ and $\angle EPQ = \angle FBQ \displaystyle$

Vertical Angles Theorem says $\displaystyle \angle EQP = \angle FQB \displaystyle$ So $\triangle EQP \sim \triangle FQB \displaystyle$ and by CPCTC $\frac {EP}{EQ} = \frac {FB}{FQ}\Longrightarrow\frac {x}{y} = \frac {1}{x}\Longrightarrow x^{2} = y \displaystyle$

Since $\displaystyle \triangle EDF$ is equilateral, $EQ + FQ = EF = BF = 1\Longrightarrow x + y = 1$. Solving for $x$ and $y$ using $\displaystyle x^{2} = y$ and $x + y = 1$ gives $\displaystyle x = \frac {\sqrt {5} - 1}{2}$ and $\displaystyle y = \frac {3 - \sqrt {5}}{2} \displaystyle$

Using the Law of Cosines, we get

$k^{2}  =  x^{2} + y^{2} - 2xy\cos{\frac {\pi}{3}}$
$=  \left(\frac {\sqrt {5} - 1}{2}\right)^{2} + \left(\frac {3 - \sqrt {5}}{2}\right)^{2} - 2\left(\frac {\sqrt {5} - 1}{2}\right)\left(\frac {3 - \sqrt {5}}{2}\right)\cos{\frac {\pi}{3}}$
$= 7 - 3\sqrt {5}$

We want the ratio of the squares of the sides, so $\displaystyle \frac {(2)^{2}}{k^{2}} = \frac {4}{7 - 3\sqrt {5}} = 7 + 3\sqrt {5}$ so $a^{2} + b^{2} + c^{2} = 7^{2} + 3^{2} + 5^{2} = 083$

See also

1998 AIME (ProblemsAnswer KeyResources)
Preceded by
Problem 11
Followed by
Problem 13
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions
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