2000 AIME I Problems/Problem 2

Revision as of 12:39, 21 October 2024 by Aeioujyot (talk | contribs) (Solutions)

Problem

Let $u$ and $v$ be integers satisfying $0 < v < u$. Let $A = (u,v)$, let $B$ be the reflection of $A$ across the line $y = x$, let $C$ be the reflection of $B$ across the y-axis, let $D$ be the reflection of $C$ across the x-axis, and let $E$ be the reflection of $D$ across the y-axis. The area of pentagon $ABCDE$ is $451$. Find $u + v$.

Solutions

Solution 1

[asy] pointpen = black; pathpen = linewidth(0.7) + black; size(180); pair A=(11,10), B=(10,11), C=(-10, 11), D=(-10, -11), E=(10, -11); D(D(MP("A\ (u,v)",A,(1,0)))--D(MP("B",B,N))--D(MP("C",C,N))--D(MP("D",D))--D(MP("E",E))--cycle); D((-15,0)--(15,0),linewidth(0.6),Arrows(5)); D((0,-15)--(0,15),linewidth(0.6),Arrows(5)); D((-15,-15)--(15,15),linewidth(0.6),Arrows(5)); [/asy]

Since $A = (u,v)$, we can find the coordinates of the other points: $B = (v,u)$, $C = (-v,u)$, $D = (-v,-u)$, $E = (v,-u)$. If we graph those points, we notice that since the latter four points are all reflected across the x/y-axis, they form a rectangle, and $ABE$ is a triangle. The area of $BCDE$ is $(2u)(2v) = 4uv$ and the area of $ABE$ is $\frac{1}{2}(2u)(u-v) = u^2 - uv$. Adding these together, we get $u^2 + 3uv = u(u+3v) = 451 = 11 \cdot 41$. Since $u,v$ are positive, $u+3v>u$, and by matching factors we get either $(u,v) = (1,150)$ or $(11,10)$. Since $v < u$ the latter case is the answer, and $u+v = \boxed{021}$.

Solution 2

We find the coordinates like in the solution above: $A = (u,v)$, $B = (v,u)$, $C = (-v,u)$, $D = (-v,-u)$, $E = (v,-u)$. Then we notice pentagon $ABCDE$ fits into a rectangle of side lengths $(u+v)$ and $(2u)$, giving us two triangles, each with hypotenuse $AB$ and $BE$. First, we can solve for the first triangle. Using the coordinates of $A$ and $B$, we discover the side lengths are both $(u-v)$, so the area of the triangle of hypotenuse $AB$ is $\frac{1}{2}(u-v)^2$. Next, we can solve for the second triangle. Using the coordinates of $A$ and $E$, we discover the side lengths are $(u-v)$ and $(u+v)$, so the area of the triangle of hypotenuse $AE$ is $\frac{1}{2}(u-v)(u+v) = \frac{1}{2}(u^2-v^2)$. Now, let’s subtract the area of these 2 triangles from the rectangle giving us $(u+v)(2u)-\frac{1}{2}(u-v)^2-\frac{1}{2}(u^2-v^2)=451 —> u^2+3uv=451 -> u(u+3v)=451$. Next, we take note of the fact that $u$ and $u+3v$ are both factors of 451, and since both $u$ and $v$ are positive integers, $u+3v$ must be greater than $u$, thus giving us two cases, where either $u=1$ or $u=11$. After trying both, the only working pair of $(u,v)$ where both $u$ and $v$ are integers are $u=11$ and $v=10$, thus meaning $u + v =$ $\boxed{021}$

$—-Aeioujyot$

Solution 3

We find the coordinates like in the solution above: $A = (u,v)$, $B = (v,u)$, $C = (-v,u)$, $D = (-v,-u)$, $E = (v,-u)$. Then we apply the Shoelace Theorem. \[A =  \frac{1}{2}[(u^2 + vu + vu + vu + v^2) - (v^2 - uv - uv - uv -u^2)] = 451\] \[\frac{1}{2}(2u^2 + 6uv) = 451\] \[u(u + 3v) = 451\]

This means that $(u,v) = (11, 10)$ or $(1,150)$, but since $v < u$, then the answer is $\boxed{021}$

See also

2000 AIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 1
Followed by
Problem 3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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