2021 AMC 12A Fall Problem 17

Attempt Problem 17 of the 2021 AMC 12A Fall 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 2021 AMC 12A Fall solutions, or check the answer key.

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17.

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?

4

6

8

12

16

“No two distinct real solutions” means each discriminant is

\le 0:

b^2 \le 4c

and

c^2 \le 4b

Multiplying gives

b^2c^2 \le 16bc,

bc \le 16;

test small positive integers

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.

Problem 17 in other years