MathDB

3

Part of 1989 Vietnam National Olympiad

Problems(2)

A square of side length 2

Source: Vietnam NMO 1989 Problem 3

2/1/2009
A square ABCD ABCD of side length 2 2 is given on a plane. The segment AB AB is moved continuously towards CD CD until A A and C C coincide with C C and D D, respectively. Let S S be the area of the region formed by the segment AB AB while moving. Prove that AB AB can be moved in such a way that S<5π6 S <\frac{5\pi}{6}.
geometrygeometry unsolved
Parallelepiped and a line

Source: Vietnam NMO 1989 Problem 6

2/1/2009
Let be given a parallelepiped ABCD.A′B′C′D′ ABCD.A'B'C'D'. Show that if a line Δ \Delta intersects three of the lines AB′ AB', BC′ BC', CD′ CD', DA′ DA', then it intersects also the fourth line.
geometry unsolvedgeometry