MathDB

2

Find the circumradius of some triangle.

ABC

is inscribed in a circle of radius 1 with centre

O;

all the angles of

ABC

are greater than

45^\circ.

B_{1}

is the foot of perpendicular from

B

to

CO,

B_{2}

is the foot of perpendicular from

B_{1}

to

AC.

Similarly,

C_{1}

is the foot of perpendicular from

C

to

BO,

C_{2}

is the foot of perpendicular from

C_{1}

to

AB.

The lines

B_{1}B_{2}

and

C_{1}C_{2}

intersect at

A_{3}.

The points

B_{3}

and

C_{3}

are defined in the same way. Find the circumradius of triangle

A_{3}B_{3}C_{3}.

Proposed by F.Bakharev, F.Petrov

Arrangement of an n x n table with entries in {0, 1}.

Zeroes and ones are arranged in all the squares of

n\times n

table. All the squares of the left column are filled by ones, and the sum of numbers in every figure of the form [asy]size(50); draw((2,1)--(0,1)--(0,2)--(2,2)--(2,0)--(1,0)--(1,2));[/asy] (consisting of a square and its neighbours from left and from below) is even. Prove that no two rows of the table are identical.
Proposed by O. Vanyushina

1

A sequence (a_n) and a polynomial P such that a_m+a_n=P(mn)?

Do there exist a sequence

a_{1}, a_{2}, a_{3}, \ldots

of real numbers and a non-constant polynomial

P(x)

such that

a_{m}+a_{n}=P(mn)

for every positive integral

m

and

n?

Proposed by A. Golovanov

4

2

The societies going against the government will be punished.

There are many opposition societies in the city of N. Each society consists of

10

members. It is known that for every

2004

societies there is a person belonging to at least

11

of them. Prove that the government can arrest

2003

people so that at least one member of each society is arrested.
Proposed by V.Dolnikov, D.Karpov

tuymaada pb

It is known that

m

and

n

are positive integers,

m > n^{n-1}

, and all the numbers

m+1

,

m+2

, \dots,

m+n

are composite. Prove that there exist such different primes

p_1

,

p_2

, \dots,

p_n

that

p_k

divides

m+k

for

k = 1

, 2, \dots,

n

.
Proposed by C. A. Grimm

2

Tuymaada pb

In the plane are given 100 lines such that no 2 are parallel and no 3 meet in a point. The intersection points are marked. Then all the lines and k of the marked points are erased. Given the remained points of intersection for what max k one can reconstruct the lines?
Proposed by A. Golovanov

Incircle, circumcircle.. one angle to compute.

The incircle of triangle

ABC

touches its sides

AB

and

BC

at points

P

and

Q.

The line

PQ

meets the circumcircle of triangle

ABC

at points

X

and

Y.

Find

\angle XBY

if

\angle ABC = 90^\circ.

Proposed by A. Smirnov

2004 Tuymaada Olympiad

Subcontests

Find the circumradius of some triangle.

Arrangement of an n x n table with entries in {0, 1}.

A sequence (a_n) and a polynomial P such that a_m+a_n=P(mn)?

The societies going against the government will be punished.

tuymaada pb

Tuymaada pb

Incircle, circumcircle.. one angle to compute.