1963 AMC 12 Problem 24

Attempt Problem 24 of the 1963 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1963 AMC 12 solutions, or check the answer key.

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24.

Consider equations of the form

x^2+bx+c=0.

How many such equations have real roots and have coefficients

b

and

c

selected from the set of integers

\{1,2,3,4,5,6\}?

20

19

18

17

16

Real roots require

b^2\geq4c

For each

c=1,\ldots,6,

count the allowed values of

b

For

c

equal to

1,

2,

3,

4,

5,

and

6,

the condition

b^2\geq4c

permits respectively

5,

4,

3,

3,

2,

and

2

values of

b

in the given set. Their sum is

5+4+3+3+2+2=19.

Thus, the correct answer is B.

Problem 24 in other years