3

Part of 2003 Tuymaada Olympiad

Problems(2)

Finite alphabet ==> finite set of words of finite lengths.

Source: Tuymaada 2003, day 1, problem 3.

A

n

S

is a set of words of finite length composed of letters of

A

. It is known that every infinite sequence of letters of

A

begins with one and only one word of

S

. Prove that the set

S

is finite.
Proposed by F. Bakharev

combinatorics proposedcombinatorics

Harmonic convex quadriteral and angles

Source: tuymaada 2003

In a convex quadrilateral

ABCD

AB\cdot CD=BC\cdot DA

2\angle A+\angle C=180^\circ

P

lies on the circumcircle of triangle

ABD

and is the midpoint of the arc

BD

A

. It is known that the point

P

lies inside the quadrilateral

ABCD

\angle BCA=\angle DCP

Proposed by S. Berlov

geometrycircumcircleangle bisectorgeometry unsolved