2

Part of 2012 CHMMC Spring

Problems(2)

2012 Spring Team #2

Source:

In the diagram below,

A

B

DE

C

A

F G

B

C

HI

DI = 5

EF = 6

GH = 7

, find the area of

\vartriangle ABC

. https://cdn.artofproblemsolving.com/attachments/d/5/90334e1bf62c99433be41f3b5e03c47c4d4916.png

2012 Spring CHMMC Tiebreaker 2 - min volume of convex octahedron

Source:

A convex octahedron in Cartesian space contains the origin in its interior. Two of its vertices are on the

x

-axis, two are on the

y

-axis, and two are on the

z

-axis. One triangular face

F

has side lengths

\sqrt{17}

\sqrt{37}

\sqrt{52}

. A second triangular face

F_0

has side lengths

\sqrt{13}

\sqrt{29}

\sqrt{34}

. What is the minimum possible volume of the octahedron?

geometry3D geometryoctahedrongeometric inequalityCHMMC