Problem:
A point is chosen at random within the square in the coordinate plane whose vertices are , and . The probability that the point is within units of a lattice point is . (A point is a lattice point if and are both integers.) What is to the nearest tenth?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
We consider an individual one-by-one block.
If we draw a quarter of a circle from each corner (where the lattice points are located), each with radius , the area covered by the circles should be . Because of this, and the fact that there are four circles, we write
Solving for , we obtain , where with , we get , and from here, we see that .
Note: To be more rigorous, note that since if then clearly the probability is greater than . This would make sure the above solution works, as if there is overlap with the quartercircles.
OR
As in the previous solution, we obtain the equation , which simplifies to . Since is slightly more than is slightly less than . We notice that is slightly more than , so is roughly
The problems on this page are the property of the MAA's American Mathematics Competitions