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Difference between revisions of "2005 AMC 12B Problems/Problem 24"

Problem

All three vertices of an equilateral triangle are on the parabola $y = x^2$, and one of its sides has a slope of $2$. The $x$-coordinates of the three vertices have a sum of $m/n$, where $m$ and $n$ are relatively prime positive integers. What is the value of $m + n$? $\mathrm{(A)}\ {{{14}}}\qquad\mathrm{(B)}\ {{{15}}}\qquad\mathrm{(C)}\ {{{16}}}\qquad\mathrm{(D)}\ {{{17}}}\qquad\mathrm{(E)}\ {{{18}}}$

Solution

Let the points be $(a,a^2)$, $(b,b^2)$ and $(c,c^2)$. Using elementary calculus concepts and the fact that they lie on $y = x^2$, $a+b$ = the slope of $AB$, $b+c$ = the slope of $BC$, $a+c$ = the slope of $AC$.

So the value that we need to find is simply the sum of the slopes of the three sides of the triangle divided by $2$. WLOG, let $AB$ be the side that has the smallest angle with the positive $x$-axis. Let $J$ be an arbitrary point with the coordinates $(1, 0)$. Let us translate the triangle so $A$ is at the origin. Then $tan(BOJ) = 2$. Using the fact that the slope of a line is equal to the tangent of the angle formed by the line and the positive x- axis, the value we now need to find is simply $\dfrac{tan(BOJ) + tan(BOJ+60) + tan(BOJ-60)}{2}$.

Using $tan(BOJ) = 2$, and basic trigonometric identities, this simplifies to $\dfrac{3}{11}$, so the answer is $3 + 11 = \boxed{\mathrm{(A)}\ 14}$

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