Question

Two parabolas have equations $y = x^2 + ax + b$ and $y = x^2 + cx + d,$ where $a,$ $b,$ $c,$ and $d$ are integers (not necessarily different), each chosen independently by rolling a fair six-sided die. What is the probability that the parabolas have at least one point in common?

$\dfrac{1}{2}$

$\dfrac{25}{36}$

$\dfrac{5}{6}$

$\dfrac{31}{36}$

$1$

Solution:

The parabolas meet where $x^2+ax+b=x^2+cx+d,$ i.e. $ax+b=cx+d.$ This has no solution exactly when the lines are parallel and distinct: $a=c$ and $b\neq d.$

The probability that $a=c$ is $\tfrac16,$ and the probability that $b\neq d$ is $\tfrac56,$ so the probability of no common point is $\tfrac16\cdot\tfrac56=\tfrac5{36}.$

The probability of at least one common point is $1-\tfrac5{36}=\tfrac{31}{36}.$

Thus, the correct answer is D.

2012 AMC 12B Problem 13

Solution:

Problem 13 in Other Years