MathDB

3

Part of 2018 Middle European Mathematical Olympiad

Problems(2)

AP ,BQ,CR are concurrent

Source: MEMO 2018 I3

9/3/2018
Let ABCABC be an acute-angled triangle with AB<AC,AB<AC, and let DD be the foot of its altitude fromA.A. Let RR and QQ be the centroids of triangles ABDABD and ACDACD, respectively. Let PP be a point on the line segment BCBC such that P≠DP \neq D and points PP QQ RR and DD are concyclic .Prove that the lines APAP BQBQ and CRCR are concurrent.
geometryptolemy sinus lemma
Pirates got killed

Source: MEMO 2018 T3

9/3/2018
A graup of pirates had an argument and not each of them holds some other two at gunpoint.All the pirates are called one by one in some order.If the called pirate is still alive , he shoots both pirates he is aiming at ( some of whom might already be dead .) All shorts are immediatcly lethal . After all the pirates have been called , it turns out the exactly 2828 pirates got killed . Prove that if the pirates were called in whatever other order , at least 1010 pirates would have been killed anyway.
combinatorics