MathDB

7

Part of 2018 Caucasus Mathematical Olympiad

Problems(2)

Interesting combinatorial problem

Source: III Caucasus Mathematical Olympiad

3/17/2018
Given a positive integer n>1n>1. In the cells of an n×nn\times n board, marbles are placed one by one. Initially there are no marbles on the board. A marble could be placed in a free cell neighboring (by side) with at least two cells which are still free. Find the greatest possible number of marbles that could be placed on the board according to these rules.
combinatorics
some angle chasing

Source: III Caucasus Mathematical Olympiad

3/17/2018
In an acute-angled triangle ABCABC, the altitudes from A,B,CA,B,C meet the sides of ABCABC at A1A_1, B1B_1, C1C_1, and meet the circumcircle of ABCABC at A2A_2, B2B_2, C2C_2, respectively. Line A1C1A_1 C_1 intersects the circumcircles of triangles AC1C2AC_1 C_2 and CA1A2CA_1 A_2 at points PP and QQ (Q≠A1Q\neq A_1, P≠C1P\neq C_1). Prove that the circle PQB1PQB_1 touches the line ACAC.
geometry