MathDB

6

Part of 2021 Saint Petersburg Mathematical Olympiad

Problems(3)

Isosceles trapezoid geo asking to prove concurrency

Source: St Petersburg 2021 11.6

12/23/2021
Point MM is the midpoint of base ADAD of an isosceles trapezoid ABCDABCD with circumcircle ω\omega. The angle bisector of ABDABD intersects ω\omega at KK. Line CMCM meets ω\omega again at NN. From point BB, tangents BP,BQBP, BQ are drawn to (KMN)(KMN). Prove that BK,MN,PQBK, MN, PQ are concurrent.
A. Kuznetsov
geometrytrapezoidcircumcircleangle bisector
Line passing through vertex of rhombus tangent to circle

Source: St Petersburg 2021 10.6

12/23/2021
A line ℓ\ell passes through vertex CC of the rhombus ABCDABCD and meets the extensions of AB,ADAB, AD at points X,YX,Y. Lines DX,BYDX, BY meet (AXY)(AXY) for the second time at P,QP,Q. Prove that the circumcircle of △PCQ\triangle PCQ is tangent to ℓ\ell
A. Kuznetsov
geometryrhombuscircumcircle
High minimum degree implies large size of matching

Source: St Petersburg 2021 9.6

12/23/2021
A school has 450450 students. Each student has at least 100100 friends among the others and among any 200200 students, there are always two that are friends. Prove that 302302 students can be sent on a kayak trip such that each of the 151151 two seater kayaks contain people who are friends.
D. Karpov
graph theorycombinatorics