<p>Consider the L-shaped region formed by three unit squares joined at their sides, as shown below. Two points <img alt="$A$" height="13" src="https://latex.artofproblemsolving.com/0/1/9/019e9892786e493964e145e7c5cf7b700314e53b.png" width="13" /> and <img alt="$B$" height="12" src="https://latex.artofproblemsolving.com/f/f/5/ff5fb3d775862e2123b007eb4373ff6cc1a34d4e.png" width="14" /> are chosen independently and uniformly at random from inside the region. The probability that the midpoint of <img alt="$\overline{AB}$" height="18" src="https://latex.artofproblemsolving.com/8/4/0/840e2b592390eb6ec918fa6f3292716ce170de66.png" width="30" /> also lies inside this L-shaped region can be expressed as <img alt="$\frac{m}{n},$" height="33" src="https://latex.artofproblemsolving.com/8/8/8/888fed89e19182447f85b3cb6676548a49395284.png" width="24" /> where <img alt="$m$" height="8" src="https://latex.artofproblemsolving.com/f/5/0/f5047d1e0cbb50ec208923a22cd517c55100fa7b.png" width="15" /> and <img alt="$n$" height="8" src="https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png" width="10" /> are relatively prime positive integers. Find <img alt="$m+n.$" height="12" src="https://latex.artofproblemsolving.com/5/d/b/5db26f75f6218a382840f0ed34f5d36bcff61d33.png" width="52" /></p> <p><img alt="[asy] unitsize(2cm); draw((0,0)--(2,0)--(2,1)--(1,1)--(1,2)--(0,2)--cycle); draw((0,1)--(1,1)--(1,0),dashed); [/asy]" height="192" src="https://latex.artofproblemsolving.com/3/5/e/35e5685ec38ac2940e2bd21b651c8faa1f022f57.png" width="192" /></p>

Question No: 396

Consider the L-shaped region formed by three unit squares joined at their sides, as shown below. Two points $A$ and $B$ are chosen independently and uniformly at random from inside the region. The probability that the midpoint of $\overline{AB}$ also lies inside this L-shaped region can be expressed as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

$[asy] unitsize(2cm); draw((0,0)--(2,0)--(2,1)--(1,1)--(1,2)--(0,2)--cycle); draw((0,1)--(1,1)--(1,0),dashed); [/asy]$

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Tags: Geometry Probability

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