MathDB

P9

Part of 2024 Bangladesh Mathematical Olympiad

Problems(2)

Product divisible by factorial

Source: BdMO 2024 Secondary National P9

3/18/2024
Find all pairs of positive integers (k,m)(k, m) such that for any positive integer nn, the product(n+m)(n+2m)⋯(n+km)(n+m)(n+2m)\cdots(n+km)is divisible by k!k!.
number theoryfactorialDivisibilityp-adic
Observe the beauty

Source: BdMO 2024 Higher Secondary National P9

3/18/2024
Let ABCABC be a triangle and MM be the midpoint of side BCBC. The perpendicular bisector of BCBC intersects the circumcircle of △ABC\triangle ABC at points KK and LL (KK and AA lie on the opposite sides of BCBC). A circle passing through LL and MM intersects AKAK at points PP and QQ (PP lies on the line segment AQAQ). LQLQ intersects the circumcircle of △KMQ\triangle KMQ again at RR. Prove that BPCRBPCR is a cyclic quadrilateral.
geometryarc midpointInversioncyclic quadrilateralSpiral Similaritypower of a point