MathDB

6

Part of 2003 AIME Problems

Problems(2)

Triangles in a Cube

Source:

5/11/2006
The sum of the areas of all triangles whose vertices are also vertices of a 1×1×11\times 1 \times 1 cube is m+n+pm+\sqrt{n}+\sqrt{p}, where mm, nn, and pp are integers. Find m+n+pm+n+p.
geometry3D geometry
Rotation about a Centroid

Source:

12/26/2006
In triangle ABC,ABC, AB=13,AB=13, BC=14,BC=14, AC=15,AC=15, and point GG is the intersection of the medians. Points A′,A', B′,B', and C′,C', are the images of A,A, B,B, and C,C, respectively, after a 180∘180^\circ rotation about G.G. What is the area if the union of the two regions enclosed by the triangles ABCABC and A′B′C′?A'B'C'?
geometrygeometric transformationrotationratioAMCAIME