Question

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?

$0.125$

$0.14$

$0.16$

$0.25$

$0.32$

Answer: C

Concepts:geometric probability lattice point area

Difficulty rating: 2340

Solution:

The diagonal from $(0.1, 0.7)$ to $(-0.1, -0.7)$ has length $\sqrt{0.2^2 + 1.4^2} = \sqrt2,$ so $S$ 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.

2012 AMC 12A Problem 23

Answer: C

Difficulty rating: 2340

Solution:

Problem 23 in Other Years