2011 AMC 12B Problem 23

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

Answer: C

Concepts:lattice point casework symmetry

Difficulty rating: 2390

Small hint:

A point

X

lies on such a path iff its taxicab distances satisfy

d(A,X)+d(X,B)\le20

Big hint:

This is

|x-3|+|x+3|

+|y-2|+|y+2|\le20;

count by symmetry across the axes

Solution:

A lattice point

X=(x,y)

lies on some path exactly when

\begin{aligned} d&=|x-3|+|x+3| \\ &\quad {}+|y-2|+|y+2|\le20. \end{aligned}

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.

If

0\le x\le3

and

0\le y\le2,

all

4\cdot3=12

points work. If

0\le x\le3

and

y\ge3,

then

y\le7,

giving

4\cdot5=20

points. If

x\ge4

and

0\le y\le2,

then

x\le8,

giving

5\cdot3=15

points. Finally, for

x\ge4

and

y\ge3,

the condition is

x+y\le10,

giving

4+3+2+1=10

points. Thus there are

57

in the first quadrant, including

15

on the nonnegative axes. By symmetry the total is

4\cdot57-2\cdot15-3=195.

Thus, the correct answer is C.

2011 AMC 12B Problem 23

Answer: C

Difficulty rating: 2390

Small hint:

Big hint:

Solution:

Problem 23 in other years