MathDB

2

Part of 2019 Czech-Polish-Slovak Junior Match

Problems(2)

3 points split a segment into 4 equal parts wanted

Source: Czech-Polish-Slovak Match Junior 2019, individual p2 CPSJ

1/20/2020
Let ABCABC be a triangle with centroid TT. Denote by MM the midpoint of BCBC. Let DD be a point on the ray opposite to the ray BABA such that AB=BDAB = BD. Similarly, let EE be a point on the ray opposite to the ray CACA such that AC=CEAC = CE. The segments TDT D and TET E intersect the side BCBC in PP and QQ, respectively. Show that the points P,QP, Q and MM split the segment BCBC into four parts of equal length.
geometryCentroidequal segments
sick rooks in a square chess board

Source: Czech-Polish-Slovak Match Junior 2019, team p2 CPSJ

1/20/2020
The chess piece sick rook can move along rows and columns as a regular rook, but at most by 22 fields. We can place sick rooks on a square board in such a way that no two of them attack each other and no field is attacked by more than one sick rook. a) Prove that on 30×3030\times 30 board, we cannot place more than 100100 sick rooks. b) Find the maximum number of sick rooks which can be placed on 8×88\times 8 board. c) Prove that on 32×3232\times 32 board, we cannot place more than 120120 sick rooks.
combinatorics