MathDB

2

Part of BIMO 2021

Problems(2)

Empty a pile of stones

Source: Own. IMO 2021 Malaysian Training Camp 1

12/31/2020
There are kk piles of stones with 20202020 stones in each pile. Amber can choose any two non-empty piles of stones, and Barbara can take one stone from one of the two chosen piles and puts it into the other pile. Amber wins if she can eventually make an empty pile. What is the least kk such that Amber can always win?
combinatorics
KM perpendicular to EF

Source: Own. IMO 2021 Malaysian Training Camp 2

1/30/2021
Let ABCABC be a triangle with incircle centered at II, tangent to sides ACAC and ABAB at EE and FF respectively. Let NN be the midpoint of major arc BACBAC. Let ININ intersect EFEF at KK, and MM be the midpoint of BCBC. Prove that KM⊥EFKM\perp EF.
geometry