MathDB

7

Part of 2004 239 Open Mathematical Olympiad

Problems(2)

200n diagonals are drawn in a convex n-gon

Source: 239MO 2004, grade 8-9, problem 8, grade 10-11, problem 7

12/11/2004
200n200n diagonals are drawn in a convex nn-gon. Prove that one of them intersects at least 10000 others. proposed by D. Karpov, S. Berlov
inductioncombinatorics unsolvedcombinatorics
PBQ passes through the circumcentre of triangle ABC

Source: 239MO 2004, grade 8-9, problem 7

12/11/2004
Given an isosceles triangle ABCABC (with AB=BCAB=BC). A point XX is chosen on a side ACAC. Some circle passes through XX, touches the side ACAC and intersects the circumcircle of triangle ABCABC in points MM and NN such that the segment MNMN bisects BXBX and intersects sides ABAB and BCBC in points PP and QQ. Prove that the circumcircle of triangle PBQPBQ passes through the circumcentre of triangle ABCABC. proposed by Sergej Berlov
geometrycircumcircleparallelogramsymmetryperpendicular bisectorgeometry solved