10

Part of 2005 AIME Problems

Problems(2)

Triangle in the Cartesian Plane

Source:

ABC

lies in the Cartesian Plane and has an area of 70. The coordinates of

B

C

(12,19)

(23,20)

, respectively, and the coordinates of

A

(p,q)

. The line containing the median to side

BC

-5

. Find the largest possible value of

p+q

geometryanalytic geometrygraphing linesslopevectorparallelogramAMC

Source:

O

is a regular octahedron, that

C

is the cube whose vertices are the centers of the faces of

O

, and that the ratio of the volume of

O

C

\frac{m}{n}

m

n

are relatively prime integers, find

m+n

geometry3D geometryoctahedronratioicosahedronpyramidtrigonometry