6

Part of 2023 India Regional Mathematical Olympiad

Problems(2)

Source: RMO 2023/6

Consider a set of

16

points arranged in

4 \times 4

square grid formation. Prove that if any

7

of these points are coloured blue, then there exists an isosceles right-angled triangle whose vertices are all blue.

CombocombinatoricsRMO 2023

Source:

AC

BD

of a cyclic quadrilateral

ABCD

P

Q

is chosen on the segment

BC

PQ

is perpendicular to

AC

. Prove that the line joining the centres of the circumcircles of triangles

APD

BQD

AD

geometrycyclic quadrilateralcircumcircle