2000 AIME II Problems/Problem 11
Contents
Problem
The coordinates of the vertices of isosceles trapezoid are all integers, with and . The trapezoid has no horizontal or vertical sides, and and are the only parallel sides. The sum of the absolute values of all possible slopes for is , where and are relatively prime positive integers. Find .
Solution
For simplicity, we translate the points so that is on the origin and . Suppose has integer coordinates; then is a vector with integer parameters (vector knowledge is not necessary for this solution). We construct the perpendicular from to , and let be the reflection of across that perpendicular. Then is a parallelogram, and . Thus, for to have integer coordinates, it suffices to let have integer coordinates.^{[1]}
Let the slope of the perpendicular be . Then the midpoint of lies on the line , so . Also, implies that . Combining these two equations yields
Since is an integer, then must be an integer. There are pairs of integers whose squares sum up to namely . We exclude the cases because they lead to degenerate trapezoids (rectangle, line segment, vertical and horizontal sides). Thus we have
These yield , and the sum of their absolute values is . The answer is
^ In other words, since is a parallelogram, the difference between the x-coordinates and the y-coordinates of and are, respectively, the difference between the x-coordinates and the y-coordinates of and . But since the latter are integers, then the former are integers also, so has integer coordinates iff has integer coordinates.
Solution 2
A very natural solution: . Shift to the origin. Suppose point was . Note is the slope we're looking for. Note that point must be of the form: or or . Note that we want the slope of the line connecting and so also be , since and are parallel. Instead of dealing with the 12 cases, we consider point of the form where we plug in the necessary values for and after simplifying. Since the slopes of and must both be , . Plugging in the possible values of in heir respective pairs and ruling out degenerate cases, we find the sum is - whatRthose
(Note: This Solution is a lot faster if you rule out due to degeneracy.)
See also
2000 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
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