1979 AMC 12 Problem 27

Question

An ordered pair

(b,c)

of integers, each of which has absolute value less than or equal to five, is chosen at random, with each such ordered pair having an equal likelihood of being chosen. What is the probability that the equation

x^2+bx+c=0

will not have distinct positive real roots?

\frac{106}{121}

\frac{108}{121}

\frac{110}{121}

\frac{112}{121}

none of these

Answer: E

Concepts:quadratic complementary probability systematic listing

Difficulty rating: 2200

Small hint:

Distinct positive roots require

b\lt0,

c\gt0,

and

b^2\gt4c

Big hint:

Count the qualifying

c

values for each

b=-1,

b=-2,

\ldots,

b=-5,

then take the complement among

121

pairs

Solution:

There are

11^2=121

ordered pairs. Distinct positive roots require

b\lt0,

c\gt0,

and

b^2\gt4c.

For

b=-1

and

b=-2

there are no choices; for

b=-3,

b=-4,

and

b=-5

there are respectively

2,

3,

and

5

choices of

c.

Thus only

10

pairs produce distinct positive roots, so the requested probability is

1-\frac{10}{121}=\frac{111}{121},

which is not listed.

Therefore, the correct answer is E.