MathDB

3

Part of 1995 Iran MO (2nd round)

Problems(2)

Find number of participants [Iran Second Round 1995]

Source:

11/25/2010
Let kk be a positive integer. 12k12k persons have participated in a party and everyone shake hands with 3k+63k+6 other persons. We know that the number of persons who shake hands with every two persons is a fixed number. Find k.k.
combinatorics proposedcombinatorics
Planes pass through a common point [ Iran Second Round 1995]

Source:

11/25/2010
In a quadrilateral ABCDABCD let A′,B′,C′A', B', C' and D′D' be the circumcenters of the triangles BCD,CDA,DABBCD, CDA, DAB and ABCABC, respectively. Denote by S(X,YZ)S(X, YZ) the plane which passes through the point XX and is perpendicular to the line YZ.YZ. Prove that if A′,B′,C′A', B', C' and D′D' don't lie in a plane, then four planes S(A,C′D′),S(B,A′D′),S(C,A′B′)S(A, C'D'), S(B, A'D'), S(C, A'B') and S(D,B′C′)S(D, B'C') pass through a common point.
geometrycircumcirclegeometry proposed