Question

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?

$4$

$6$

$8$

$12$

$16$

Answer: B

Concepts:quadratic inequality

Difficulty rating: 1910

Solution:

Neither quadratic has two distinct real roots exactly when both discriminants are nonpositive: $b^2 \le 4c$ and $c^2 \le 4b.$

Combining $c\ge b^2/4$ with $c\le2\sqrt b$ gives $b^{3/2}\le8,$ so $b\le4.$ 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.

2021 AMC 12A Fall Problem 17

Answer: B

Difficulty rating: 1910

Solution:

Problem 17 in Other Years