MathDB

3

Part of 2012 Saint Petersburg Mathematical Olympiad

Problems(3)

Pyramid base

Source: St Petersburg Olympiad 2012, Grade 11, P3

9/29/2017
At the base of the pyramid SABCDSABCD lies a convex quadrilateral ABCDABCD, such that BC∗AD=BD∗ACBC * AD = BD * AC. Also ∠ADS=∠BDS,∠ACS=∠BCS \angle ADS =\angle BDS ,\angle ACS =\angle BCS. Prove that the plane SABSAB is perpendicular to the plane of the base.
geometry3D geometrypyramid
Questions of exam.

Source: St Petersburg Olympiad 2012, Grade 10, P3

9/29/2017
2525 students are on exams. Exam consists of some questions with 55 variants of answer. Every two students gave same answer for not more than 11 question. Prove, that there are not more than 66 questions in exam.
combinatorics
Some geometry

Source: St Petersburg Olympiad 2012, Grade 9, P3

9/29/2017
ABCDABCD is inscribed. Bisector of angle between diagonals intersect ABAB anc CDCD at XX and YY. M,NM,N are midpoints of AD,BCAD,BC. XM=YMXM=YM Prove, that XN=YNXN=YN.
geometry