MathDB

4

Part of 2021 Caucasus Mathematical Olympiad

Problems(2)

Another problem on a square grid

Source: VI Caucasus Mathematical Olympiad

3/14/2021
A square grid 2n×2n2n \times 2n is constructed of matches (each match is a segment of length 1). By one move Peter can choose a vertex which (at this moment) is the endpoint of 3 or 4 matches and delete two matches whose union is a segment of length 2. Find the least possible number of matches that could remain after a number of Peter's moves.
combinatorics
The reflections of the vertices and the intersections of circumcircle with HaHb

Source: VI Caucasus Mathematical Olympiad

3/14/2021
In an acute triangle ABCABC let AHaAH_a and BHbBH_b be altitudes. Let HaHbH_aH_b intersect the circumcircle of ABCABC at PP and QQ. Let A′A' be the reflection of AA in BCBC, and let B′B' be the reflection of BB in CACA. Prove that A′,B′A', B', PP, QQ are concyclic.
geometrygeometric transformationreflectioncircumcircle