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?

$0.125$

$0.14$

$0.16$

$0.25$

$0.32$

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.

The overlap of two unit-area copies whose centers are one unit apart has area $0.08.$ Summing over the horizontal and vertical adjacencies gives probability $2 \cdot 0.08 = 0.16.$

Thus, the correct answer is C.

2012 AMC 12A Problem 23

Solution:

Problem 23 in Other Years