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Difference between revisions of "Mock AIME 1 2006-2007 Problems/Problem 1"

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==Problem==
 
<math>\triangle ABC</math> has positive integer side lengths of <math>x</math>,<math>y</math>, and <math>17</math>.  The angle bisector of <math>\angle BAC</math> hits <math>BC</math> at <math>D</math>. If <math>\angle C=90^\circ</math>, and the maximum value of <math>\frac{[ABD]}{[ACD]}=\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive intgers, find <math>m+n</math>. (Note <math>[ABC]</math> denotes the area of <math>\triangle ABC</math>).
 
<math>\triangle ABC</math> has positive integer side lengths of <math>x</math>,<math>y</math>, and <math>17</math>.  The angle bisector of <math>\angle BAC</math> hits <math>BC</math> at <math>D</math>. If <math>\angle C=90^\circ</math>, and the maximum value of <math>\frac{[ABD]}{[ACD]}=\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive intgers, find <math>m+n</math>. (Note <math>[ABC]</math> denotes the area of <math>\triangle ABC</math>).
  

Revision as of 16:08, 23 August 2006

Problem

$\triangle ABC$ has positive integer side lengths of $x$,$y$, and $17$. The angle bisector of $\angle BAC$ hits $BC$ at $D$. If $\angle C=90^\circ$, and the maximum value of $\frac{[ABD]}{[ACD]}=\frac{m}{n}$ where $m$ and $n$ are relatively prime positive intgers, find $m+n$. (Note $[ABC]$ denotes the area of $\triangle ABC$).

Solution

Assume without loss of generality that $x \leq y$. Then the hypotenuse of right triangle $\triangle ABC$ either has length 17, in which case $x^2 + y^2 = 17$, or has length $y$, in which case $x^2 + 17^2 = y^2$, by the Pythagorean Theorem.

In the first case, you can either know your Pythagorean triples or do a bit of casework to find that the only solution is $x = 8, y = 15$. In the second case, we have $17^2 = y^2 - x^2 = (y - x)(y + x)$, a factorization as a product of two different positive integers, so we must have $y - x = 1$ and $y + x = 17^2 = 289$ from which we get the solution $x = 144, y= 145$.

Now, note that the area $[ABD] = \frac 12 AB \cdot AD \cdot \sin BAD$ and $[ACD] = \frac 12 \cdot AD \cdot AC \cdot \sin CAD$, and since $AD$ is an angle bisector we have $\displaystyle \angle BAD = \angle CAD$ so $\frac{[ABD]}{[ACD]} = \frac{AB}{AC}$.

In our first case, this value may be either $\frac {17}{8}$ or $\frac{17}{15}$. In the second, it may be either $\frac{145}{144}$ or $\frac{145}{17}$. Of these four values, the last is clearly the greatest. 17 and 145 are relatively prime, so our answer is $17 + 145 = 162$.


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