MathDB

A7

Part of 2012 Princeton University Math Competition

Problems(2)

2012 PUMaC Geometry A7

Source:

9/24/2019
An octahedron (a solid with 8 triangular faces) has a volume of 10401040. Two of the spatial diagonals intersect, and their plane of intersection contains four edges that form a cyclic quadrilateral. The third spatial diagonal is perpendicularly bisected by this plane and intersects the plane at the circumcenter of the cyclic quadrilateral. Given that the side lengths of the cyclic quadrilateral are 7,15,24,207, 15, 24, 20, in counterclockwise order, the sum of the edge lengths of the entire octahedron can be written in simplest form as a/ba/b. Find a+ba + b.
geometry3D geometry
2012 PUMaC Number Theory A7

Source:

10/5/2019
Let a,ba, b, and cc be positive integers satisfying a4+a2b2+b4=9633a^4 + a^2b^2 + b^4 = 9633 2a2+a2b2+2b2+c5=36052a^2 + a^2b^2 + 2b^2 + c^5 = 3605. What is the sum of all distinct values of a+b+ca + b + c?
number theory