Hypocycloid and Lattice points
Problem 450
A hypocycloid is the curve drawn by a point on a small circle rolling inside a larger circle. The parametric equations of a hypocycloid centered at the origin, and starting at the right most point is given by:
Where R is the radius of the large circle and r the radius of the small circle.
Let be the set of distinct points with integer coordinates on the hypocycloid with radius R and r and for which there is a corresponding value of t such that and are rational numbers.
Let be the sum of the absolute values of the x and y coordinates of the points in .
Let be the sum of for R and r positive integers, and .
You are given:
C(3, 1) = C(2500, 1000) = Note: (-625, 0) is not an element of C(2500, 1000) because is not a rational number for the corresponding values of t. S(3, 1) = (|3| + |0|) + (|-1| + |2|) + (|-1| + |0|) + (|-1| + |-2|) = 10 T(3) = 10; T(10) = 524 ;T(100) = 580442; T(103) = 583108600. Find T(106). |