Question

A bug travels in the coordinate plane, moving only along the lines that are parallel to the $x$ -axis or $y$ -axis. Let $A=(-3, 2)$ and $B=(3, -2).$ Consider all possible paths of the bug from $A$ to $B$ of length at most $20.$ How many points with integer coordinates lie on at least one of these paths?

$161$

$185$

$195$

$227$

$255$

Solution:

A lattice point $X=(x,y)$ lies on some path exactly when $d=|x-3|+|x+3|+|y-2|+|y+2|\le20.$ This expression is unchanged when $x\to-x$ or $y\to-y,$ so we count points with $x\ge0,$ $y\ge0,$ multiply by $4,$ and correct for the axes.

Splitting into the four regions determined by whether $x\le3$ and $y\le2$ gives $12+20+15+10=57$ points in the first quadrant (including axis points). By symmetry the total is $4\cdot57-2\cdot15-3=195.$

Thus, the correct answer is C.

2011 AMC 12B Problem 23

Solution:

Problem 23 in Other Years