MathDB

5

Part of 2018 India Regional Mathematical Olympiad

Problems(2)

Triangular Numbers in action

Source: RMO 2018 P5

10/7/2018
Find all natural numbers nn such that 1+[2n] 1+[\sqrt{2n}]~ divides 2n2n.
( For any real number xx , [x][x] denotes the largest integer not exceeding xx. )
algebranumber theoryDivisibility
Angle chasing yet again

Source: RMO 2018 P5

10/28/2018
In a cyclic quadrilateral ABCDABCD with circumcenter OO, the diagonals ACAC and BDBD intersect at XX. Let the circumcircles of triangles AXDAXD and BXCBXC intersect at YY. Let the circumcircles of triangles AXBAXB and CXDCXD intersect at ZZ. If OO lies inside ABCDABCD and if the points O,X,Y,ZO,X,Y,Z are all distinct, prove that O,X,Y,ZO,X,Y,Z lie on a circle.
geometrycyclic quadrilateralcircumcircle