2021 Fall AMC 12A Problem 17

Problem 17 of 25IntermediateAlgebra

For how many ordered pairs

(b, c)

of positive integers does neither

x^2 + bx + c = 0

nor

x^2 + cx + b = 0

have two distinct real solutions?

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Solution

Neither quadratic has two distinct real roots exactly when both discriminants are nonpositive:

b^2 \le 4c

and

c^2 \le 4b.

Combining

c\ge \frac{b^2}{4}

with

c\le2\sqrt b

gives

b^{\frac{3}{2}}\le8,

b\le4.

Checking:

b = 1

gives

c \in \{1,2\};

b = 2

gives

c \in \{1,2\};

b = 3

gives

c = 3;

and

b = 4

gives

c = 4.

That is

(1,1),

(1,2),

(2,1),

(2,2),

(3,3),

(4,4)

—

6

ordered pairs. Thus, the correct answer is B.