MathDB

2

Part of 2020 Iran Team Selection Test

Problems(2)

A crowded geometry

Source: Iran TST1 Day1 P2

2/22/2020
Let OO be the circumcenter of the triangle ABCABC. Points D,ED,E are on sides AC,ABAC,AB and points P,Q,R,SP,Q,R,S are given in plane such that P,CP,C and R,CR,C are on different sides of ABAB and pints Q,BQ,B and S,BS,B are on different sides of ACAC such that R,SR,S lie on circumcircle of DAP,EAQDAP,EAQ and △BCE∼△ADQ,△CBD∼△AEP\triangle BCE \sim \triangle ADQ , \triangle CBD \sim \triangle AEP(In that order), ∠ARE=∠ASD=∠BAC\angle ARE=\angle ASD=\angle BAC, If RS∥PQRS\| PQ prove that RE,DSRE ,DS are concurrent on AOAO.
Proposed by Alireza Dadgarnia
geometryIranian TSTcircumcircle
A game on a board

Source: Iran TST2 Day1 P2

3/11/2020
Alice and Bob take turns alternatively on a 2020×20202020\times2020 board with Alice starting the game. In every move each person colours a cell that have not been coloured yet and will be rewarded with as many points as the coloured cells in the same row and column. When the table is coloured completely, the points determine the winner. Who has the wining strategy and what is the maximum difference he/she can grantees?
Proposed by Seyed Reza Hosseini
combinatorics2020game