2007 iTest Problems/Problem 32

Problem

When a rectangle frames a parabola such that a side of the rectangle is parallel to the parabola's axis of symmetry, the parabola divides the rectangle into regions whose areas are in the ratio $2$ to $1$ . How many integer values of k are there such that $0<k\leq 2007$ and the area between the parabola $y=k-x^2$ and the $x$ -axis is an integer?

$[asy] import graph; size(300); defaultpen(linewidth(0.8)+fontsize(10)); real k=1.5; real endp=sqrt(k); real f(real x) { return k-x^2; } path parabola=graph(f,-endp,endp)--cycle; filldraw(parabola, lightgray); draw((endp,0)--(endp,k)--(-endp,k)--(-endp,0)); label("Region I", (0,2*k/5)); label("Box II", (51/64*endp,13/16*k)); label("area(I) = $\frac23$\,area(II)",(5/3*endp,k/2));[/asy]$

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2007 iTest Problems/Problem 32

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