2013 AMC 12B Problem 21

Problem 21 of 25HarderGeometry Combinatorics

Consider the set of

30

parabolas defined as follows: all parabolas have as focus the point

(0, 0)

and the directrix lines have the form

y = ax + b

with

a

and

b

integers such that

a \in \{-2, -1, 0, 1, 2\}

and

b \in \{-3, -2, -1, 1, 2, 3\}.

No three of these parabolas have a common point. How many points in the plane are on two of these parabolas?

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Solution

Two parabolas with common focus

O

meet in exactly

2

points, except when their directrices are parallel and

O

lies outside the strip between them, in which case they do not meet. The non-intersecting pairs have directrices of equal slope and

y

-intercepts of the same sign. There are

5

slopes, and for each,

2\binom{3}{2} = 6

same-sign intercept pairs. Since every intersecting pair meets in

2

points and no point lies on three parabolas, the total is

2\left(\binom{30}{2} - 5\cdot 6\right)

= 2(435 - 30)

= 810.

Thus, the correct answer is C.

Tagged: parabola · complementary counting · combinations