Math Gold Medalist

Lor

2023 AIME II 

Problem 6

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]


Changing Probability to Geometry

Coordinate Geometry

Consider A(x1, y1) and B(x2, y2) then the midpoint M((x1+x2)/2, (y1+y2)/2)

Calculate P(x1+x2>2) and P(y1+y2>2)

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   Solution