Problem

2006 AIME I Problem 10

Eight circles of diameter 1 are packed in the first quadrant of the coordinate plane as shown. Let region \mathcal{R} be the union of the eight circular regions. Line l, with slope 3, divides \mathcal{R} into two regions of equal area. Line l's equation can be expressed in the form ax=by+c, where a, b, and c are positive integers whose greatest common divisor is 1. Find a^2+b^2+c^2.

[asy] unitsize(0.50cm); draw((0,-1)--(0,6)); draw((-1,0)--(6,0)); draw(shift(1,1)*unitcircle); draw(shift(1,3)*unitcircle); draw(shift(1,5)*unitcircle); draw(shift(3,1)*unitcircle); draw(shift(3,3)*unitcircle); draw(shift(3,5)*unitcircle); draw(shift(5,1)*unitcircle); draw(shift(5,3)*unitcircle); [/asy]

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.


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Category: AIME I
Points: 5
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