3

Part of 1985 Vietnam National Olympiad

Problems(2)

Calculate the area

Source: Vietnam NMO 1985 Problem 3

A parallelepiped with the side lengths

a

b

c

is cut by a plane through its intersection of diagonals which is perpendicular to one of these diagonals. Calculate the area of the intersection of the plane and the parallelepiped.

geometrygeometry unsolved

calculate the volume of the pyramid

Source: Vietnam NMO 1985 Problem 6

A triangular pyramid

O.ABC

ABC

has the property that the lengths of the altitudes from

A

B

C

are not less than \frac{OB \plus{}OC}{2}, \frac{OC \plus{} OA}{2} and \frac{OA \plus{} OB}{2}, respectively. Given that the area of

ABC

S

, calculate the volume of the pyramid.

geometry3D geometrypyramidgeometry unsolved