2000 AIME II Problems/Problem 11: Difference between revisions

Revision as of 22:15, 10 March 2010

Problem

The coordinates of the vertices of isosceles trapezoid $ABCD$ are all integers, with $A=(20,100)$ and $D=(21,107)$ . The trapezoid has no horizontal or vertical sides, and $\overline{AB}$ and $\overline{CD}$ are the only parallel sides. The sum of the absolute values of all possible slopes for $\overline{AB}$ is $m/n$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

Solution

For simplicity, we translate the points so that $A$ is on the origin and $D = (1,7)$ . Suppose $B$ has integer coordinates; then $\overrightarrow{AB}$ is a vector with integer parameters (vector knowledge is not necessary for this solution). We construct the perpendicular from $A$ to $\overline{CD}$ , and let $D' = (a,b)$ be the reflection of $D$ across that perpendicular. Then $ABCD'$ is a parallelogram, and $\overrightarrow{AB} = \overrightarrow{D'C}$ . Thus, for $C$ to have integer coordinates, it suffices to let $D'$ have integer coordinates.^[1]

$[asy] pathpen = linewidth(0.7); pair A=(0,0), D=(1,7), Da = MP("D'",D((-7,1)),N), B=(-8,-6), C=B+Da, F=foot(A,C,D); D(MP("A",A)--MP("B",B)--MP("C",C,N)--MP("D",D,N)--cycle); D(F--A--Da,linetype("4 4")); [/asy]$

Let the slope of the perpendicular be $m$ . Then the midpoint of $\overline{DD'}$ lies on the line $y=mx$ , so $\frac{b+7}{2} = m \cdot \frac{a+1}{2}$ . Also, $AD = AD'$ implies that $a^2 + b^2 = 1^2 + 7^2 = 50$ . Combining these two equations yields

$a^2 + \left(7 - (a+1)m\right)^2 = 50$

Since $a$ is an integer, then $7-(a+1)m$ must be an integer. There are $12$ pairs of integers whose squares sum up to $50,$ namely $( \pm 1, \pm 7), (\pm 7, \pm 1), (\pm 5, \pm 5)$ . We exclude the cases $(\pm 1, \pm 7)$ because they lead to degenerate trapezoids (rectangle, line segment, vertical and horizontal sides). Thus we have

$7 - 8m = \pm 1, \quad 7 + 6m = \pm 1, \quad 7 - 6m = \pm 5, 7 + 4m = \pm 5$

These yield $m = 1, \frac 34, -1, -\frac 43, 2, \frac 13, -3, - \frac 12$ , and the sum of their absolute values is $\frac{119}{12}$ . The answer is $m+n= \boxed{131}$

^ In other words, since $ABCD'$ is a parallelogram, the difference between the x-coordinates and the y-coordinates of $C$ and $D'$ are, respectively, the difference between the x-coordinates and the y-coordinates of $A$ and $B$ . But since the latter are integers, then the former are integers also, so $C$ has integer coordinates iff $D'$ has integer coordinates.

@@ Line 15: / Line 15: @@
 <cmath>a^2 + \left(7 - (a+1)m\right)^2 = 50</cmath>
-Since <math>a</math> is an integer, then <math>7-(a+1)m</math> must be an integer. There are <math>12</math> pairs of integers whose squares sum up to <math>50</math>, namely <math>( \pm 1, \pm 7), (\pm 7, \pm 1), (\pm 5, \pm 5)</math>. We exclude the cases <math>(\pm 1, \pm 7)</math> because they lead to degenerate trapezoids (rectangle, line segment, vertical and horizontal sides). Thus we have
+Since <math>a</math> is an integer, then <math>7-(a+1)m</math> must be an integer. There are <math>12</math> pairs of integers whose squares sum up to <math>50,</math> namely <math>( \pm 1, \pm 7), (\pm 7, \pm 1), (\pm 5, \pm 5)</math>. We exclude the cases <math>(\pm 1, \pm 7)</math> because they lead to degenerate trapezoids (rectangle, line segment, vertical and horizontal sides). Thus we have
 <cmath>7 - 8m = \pm 1, \quad 7 + 6m = \pm 1, \quad 7 - 6m = \pm 5, 7 + 4m = \pm 5</cmath>

2000 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 10	Followed by Problem 12
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2000 AIME II Problems/Problem 11: Difference between revisions

Revision as of 22:15, 10 March 2010

Problem

Solution

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