MathDB

7

Part of 2010 Math Prize For Girls Problems

Problems(1)

Math Prize 2010 Problem 7

Source:

11/14/2010
The graph of (x2+y2āˆ’1)3=x2y3{(x^2 + y^2 - 1)}^3 = x^2 y^3 is a heart-shaped curve, shown in the figure below. [asy] import graph; unitsize(10); real f(real x) { return sqrt(cbrt(x^4) - 4 x^2 + 4); } real g(real x) { return (cbrt(x^2) + f(x))/2; } real h(real x) { return (cbrt(x^2) - f(x)) / 2; } real xmax = 1.139028; draw(graph(g, -xmax, xmax) -- reverse(graph(h, -xmax, xmax)) -- cycle); xaxis("xx", -1.5, 1.5, above = true); yaxis("yy", -1.5, 1.5, above = true); [/asy] For how many ordered pairs of integers (x,y)(x, y) is the point (x,y)(x, y) inside or on this curve?
analytic geometry