3

Part of 2007 All-Russian Olympiad

Problems(2)

Source: All-Russian 2007

Given a rhombus

ABCD

M

is chosen on its side

BC

. The lines, which pass through

M

and are perpendicular to

BD

AC

AD

P

Q

respectively. Suppose that the lines

PB,QC,AM

have a common point. Find all possible values of a ratio

\frac{BM}{MC}

. S. Berlov, F. Petrov, A. Akopyan

geometryrhombusratiogeometry unsolved

draw diagonals in a regular (2n+1)-gon

Source: All-Russian 2007

Two players by turns draw diagonals in a regular

(2n+1)

n>1

). It is forbidden to draw a diagonal, which was already drawn, or intersects an odd number of already drawn diagonals. The player, who has no legal move, loses. Who has a winning strategy? K. Sukhov

geometrygeometric transformationreflectioncombinatorics unsolvedcombinatorics