Crisscross Ellipses

$E_a$ is an ellipse with an equation of the form $x^2 + 4y^2 = 4a^2$.
$E_a^\prime$ is the rotated image of $E_a$ by $\theta$ degrees counterclockwise around the origin $O(0, 0)$ for $0^\circ \lt \theta \lt 90^\circ$.

$b$ is the distance to the origin of the two intersection points closest to the origin and $c$ is the distance of the two other intersection points.
We call an ordered triplet $(a, b, c)$ a canonical ellipsoidal triplet if $a, b$ and $c$ are positive integers.
For example, $(209, 247, 286)$ is a canonical ellipsoidal triplet.

Let $C(N)$ be the number of distinct canonical ellipsoidal triplets $(a, b, c)$ for $a \leq N$.
It can be verified that $C(10^3) = 7$, $C(10^4) = 106$ and $C(10^6) = 11845$.

Find $C(10^{17})$.

This problem seems to arise from a higher level competitive math exam, such as those from the International Mathematical Olympiad (IMO). However, let’s break it down and see what we can figure out.

First off, because of the complexity of the question and the huge bounds of the function C(N), we must use a program to compute C(N) for N as large as 10^17. The problem inherently involves geometric transformations, specifically rotations of ellipses, which in turn require computation of intersection points and distances.

Without a specific algorithm for this kind of problem, we can guess that a brute-force approach would involve running through all a <= N, calculating the rotated ellipses, finding the intersection points, and then checking if the distances b and c are positive integers. Here are some steps we might use to implement a solution in principle (though actually programming this solution would be an enormous task, especially considering the very large value of N = 10^17): 1. Iterate over all a in the range 1 - N inclusive. 2. For each a, derive the equation of the ellipse E_a and E_a prime. 3. Calculate the solutions of the system composed by the equations of E_a and E_a prime (these correspond to the intersections of the rotated ellipses). 4. From the solutions obtained, compute the distances b and c. 5. Check if b and c are positive integers. If they are, increase the counter for C(N). 6. Continue until you have checked all a in the range 1 - N. Again, this approach without further optimizations, would likely not run in a reasonable amount of time for large values of N, like N=10^17, because of the complexity of the computations involved. This problem likely requires more advanced techniques related to elliptic curves and algebraic geometry for an efficient solution. Currently, it seems too complicated and might require an advanced mathematical background to solve. It would be recommended to consult with an expert in that field or find the solution of this problem from IMO resources.

More Answers: