1976 AMC 12 Problem 8

Question

A point in the plane, both of whose rectangular coordinates are integers with absolute value less than or equal to four, is chosen at random, with all such points having an equal probability of being chosen. What is the probability that the distance from the point to the origin is at most two units?

\frac{13}{81}

\frac{15}{81}

\frac{13}{64}

\frac{\pi}{16}

the square of a rational number

Answer: A

Concepts:lattice point basic probability circle systematic listing

Difficulty rating: 1710

Small hint:

There are

9

choices for each coordinate

Big hint:

Count integer pairs satisfying

x^2+y^2\le4

by considering

x=0,

x=\pm1,

and

x=\pm2

Solution:

There are

9^2=81

possible points. Within distance

2

are the origin; the four points

(\pm1,0),

(0,\pm1);

the four points

(\pm1,\pm1);

and the four points

(\pm2,0),

(0,\pm2).

Thus

13

points qualify, for probability

\frac{13}{81}.

Therefore, the correct answer is A.