MathDB

2

Part of 2000 Vietnam National Olympiad

Problems(2)

Vietnam NMO 2000_2

Source:

10/26/2008
Two circles (O1) (O_1) and (O2) (O_2) with respective centers O1 O_1, O2 O_2 are given on a plane. Let M1 M_1, M2 M_2 be points on (O1) (O_1), (O2) (O_2) respectively, and let the lines O1M1 O_1M_1 and O2M2 O_2M_2 meet at Q Q. Starting simultaneously from these positions, the points M1 M_1 and M2 M_2 move clockwise on their own circles with the same angular velocity. (a) Determine the locus of the midpoint of M1M2 M_1M_2. (b) Prove that the circumcircle of △M1QM2 \triangle M_1QM_2 passes through a fixed point.
geometrycircumcirclegeometric transformationgeometry proposed
Vietnam NMO 2000_5

Source:

10/26/2008
Find all integers n≥3 n \ge 3 such that there are n n points in space, with no three on a line and no four on a circle, such that all the circles pass through three points between them are congruent.
geometry unsolvedgeometry