2012 AMC 12A Problem 23

Problem 23 of 25HarderGeometry Counting & Probability

Let

S

be the square one of whose diagonals has endpoints

(0.1, 0.7)

and

(-0.1, -0.7).

A point

v = (x, y)

is chosen uniformly at random over all pairs of real numbers

x

and

y

such that

0 \le x \le 2012

and

0 \le y \le 2012.

Let

T(v)

be a translated copy of

S

centered at

v.

What is the probability that the square region determined by

T(v)

contains exactly two points with integer coordinates in its interior?

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Solution

The diagonal from

(0.1, 0.7)

(-0.1, -0.7)

has length

\sqrt{0.2^2 + 1.4^2} = \sqrt2,

S

is a square of area

1.

The translate

T(v)

contains a lattice point exactly when

v

lies inside the copy of

S

centered at that point. Containing exactly two interior lattice points requires

v

to lie in the overlap of two copies centered at adjacent lattice points. By periodicity the answer is the total such overlap area within one unit cell. Consider copies centered at

(0,0)

and

(1,0).

Their overlap is a rectangle. A quarter-turn of the listed half-diagonal gives the relevant vertices

(0.7,-0.1)

and

(0.3,0.1).

Following the two pairs of parallel sides through their intersections gives side lengths

0.4

and

0.2,

so the overlap area is

0.08.

Diagonally centered copies do not overlap, and horizontal and vertical adjacencies contribute equally. Accounting for the cell boundaries gives probability

2\cdot0.08=0.16.

Thus, the correct answer is C.

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