MathDB

1

Maximal number of dominos can be cut from a polygon

1\times20

,

1\times 19

, ...,

1\times 1

were cut out of

20\times20

table. Prove that the maximal amount of

1\times2

dominos can be cut from the table is 85
Proposed by S. Berlov

11.6

1

Integer FE from St. Petersburg

Find all functions

f:\mathbb{Z}\rightarrow\mathbb{Z}

such that for any

x,y

the following is true:

f(x+y+f(y))=f(x)+2y

[I]proposed by F. Petrov

11.5

1

$MI=MH$, where M is the midpoint of arc $AC$

Let

I

and

H

be the incenter and orthocenter of an acute triangle

ABC

.

M

is the midpoint of arc

AC

of circumcircle of triangle

ABC

which does not contain point

B

. If

MI=MH

, find the measure of angle

\angle ABC

.
[I]Proposed by F. Bakharev

11.4

1

Inequality with gcd and square root

For any two positive integers

n>m

prove the following inequality:

[m,n]+[m+1,n+1]\geq \dfrac{2nm}{\sqrt{m-n}}

As always,

[x,y]

means the least common multiply of

x,y

.
[I]Proposed by A. Golovanov

11.2

1

Algorithms on graphs

There are 2000 cities in a country and no roads. Prove that some cities can be connected by a road such that there would be 2 cities with 1 road passing through them, there would be 2 cities with 2 roads passim through them,...,there would be 2 cities with 1000 roads passing through them.
[I]Proposed by F. Bakharev

11.1

1

Algebra on trigonometry

Do there exist distinct numbers

x,y,z

from

[0,\dfrac{\pi}{2}]

, such that six number

\sin x

,

\sin y

,

\sin z

,

\cos x

,

\cos y

,

\cos z

could be partitioned into 3 pairs with equal sums?
[I]Proposed by A. Golovanov

10.7

1

Graph theory and processes

On the parliament of Sikinia, for any two deputies, there is third deputy, which knows exactly one of the two. Every deputy belongs to one of the two ruling parties. Every day, he president tells a certain group of deputies to change the party that they belong, and all the deputies which which know at least one of the deputies of the group has to change their party too. Prove that, the president can reach any configuration of deputies between two parties.(The president himself isn't a member of the parliament of Sikinia).
[I]Proposed by S. Berlov

10.6

1

Inequality with gcd

For any positive integers

n>m

prove the following inequality:

[m,n]+[m+1,n+1]\geq 2m\sqrt{n}

As usual, [x,y] denotes the least common multiply of

x,y

[I]Proposed by A. Golovanov

10.5

1

Find angle AMN

On the bisector

AL

of triangle

ABC

a point

K

is chosen such that

\angle BKL=\angle KBL=30^{\circ}

. Lines

AB

and

CK

intersect at point

M

, lines

AC

and

BK

intersect at point

N

. FInd the measure of angle

\angle AMN

[I]Proposed by D. Shiryaev, S. Berlov

10.4

1

Prove that a polygon can be divided into dominos

Rectangles

1\times20

,

1\times 19

, ...,

1\times 1

were cut out of

20\times20

table. Prove that from the remaining part of the table

36

1\times2

dominos can be cut
[I]Proposed by S. Berlov

10.3

1

AB+BC=3AC

Let

I

be the incenter of triangle

ABC

and let

D

be the midpoint of side

AB

. Prove that if the angle

\angle AOD

is right, then

AB+BC=3AC

.
[I]Proposed by S. Ivanov

10.2

1

A certain operation with numbers

The computer "Intel stump-V" can do only one operation with a number: add 1 to it, then rearrange all the zeros in the decimal representation to the end and rearrenge the left digits in any order. (For example from 1004 you could get 1500 or 5100). The number

12345

was written on the computer and after performing 400 operations, the number 100000 appeared on the screen. How many times has a number with the last digit 0 appeared on the screen?

10.1

1

Inequality on quadratic trinomials

Quadratic trinomials

f

and

g

with integer coefficients obtain only positive values and the inequality

\dfrac{f(x)}{g(x)}\geq \sqrt{2}

is true

\forall x\in\mathbb{R}

. Prove that

\dfrac{f(x)}{g(x)}>\sqrt{2}

is true

\forall x\in\mathbb{R}

[I]Proposed by A. Khrabrov

9.6

1

Diophanative with exponents

Find all positive integer solution:

k^m+m^n=k^n+1

[I]Proposed by V. Frank, F. Petrov

9.7

1

Distributing on a circular table

300 students participate on the international math olympiad. Every student speaks in exactly two of the official languages of the olympiad and every language is spoken by 100 people (it is known that students speak only the official languages). Prove that the students can be sited on a circular table, such that no two neighbors spoke the same language.

9.5

1

Beautiful problem on midlines and isogonal

Points

A_1

,

B_1

,

C_1

are midpoints of sides

BC

,

AC

,

AB

of triangle

ABC

. On midlines

C_1B_1

and

A_1B_1

points

E

and

F

are chosen such that

BE

is the angle bisector of

AEB_1

and

BF

is the angle bisector of

CFB_1

. Prove that bisectors of angles

ABC

and

FBE

coincide.
[I]Proposed by F. Baharev

9.1

1

Game on coloring cells of a board.

All the cells of a

10\times10

board are colored white initially. Two players are playing a game with alternating moves. A move consists of coloring any un-colored cell black. A player is considered to loose, if after his move no white domino is left. Which of the players has a winning strategy?
[I]Proposed by A. Khrabrov

9.2

1

10 numbers forming 500 "good" quadratic trinomials

Define a quadratic trinomial to be "good", if it has two distinct real roots and all of its coefficients are distinct. Do there exist 10 positive integers such that there exist 500 good quadratic trinomials coefficients of which are among these numbers?
[I]Proposed by F. Petrov

9.4

1

Prove that (ab+c, bc+a, ca+b)=(a,b,c)

Let

a,b,c\in\mathbb{Z^{+}}

such that

(a^2-1, b^2-1, c^2-1)=1

Prove that

(ab+c, bc+a, ca+b)=(a,b,c)

(As usual,

(x,y,z)

means the greatest common divisor of numbers

x,y,z

) [I]Proposed by A. Golovanov

9.3

1

On a circumscribed pentagon

A convex pentagon

ABCDE

is given with

AB=BC

,

CD=DE

and

\angle A=\angle C=\angle E>90^{\circ}

. Prove that the pentagon is circumscribed
[I]Proposed by F. Baharev

2001 Saint Petersburg Mathematical Olympiad

Subcontests

Maximal number of dominos can be cut from a polygon

Integer FE from St. Petersburg

$MI=MH$, where M is the midpoint of arc $AC$

Inequality with gcd and square root

Algorithms on graphs

Algebra on trigonometry

Graph theory and processes

Inequality with gcd

Find angle AMN

Prove that a polygon can be divided into dominos

AB+BC=3AC

A certain operation with numbers

Inequality on quadratic trinomials

Diophanative with exponents

Distributing on a circular table

Beautiful problem on midlines and isogonal

Game on coloring cells of a board.

10 numbers forming 500 "good" quadratic trinomials

Prove that (ab+c, bc+a, ca+b)=(a,b,c)

On a circumscribed pentagon

2001 Saint Petersburg Mathematical Olympiad

Subcontests

Maximal number of dominos can be cut from a polygon

Integer FE from St. Petersburg

$MI=MH$, where M is the midpoint of arc $AC$

Inequality with gcd and square root

Algorithms on graphs

Algebra on trigonometry

Graph theory and processes

Inequality with gcd

Find angle AMN

Prove that a polygon can be divided into dominos

AB+BC=3AC

A certain operation with numbers

Inequality on quadratic trinomials

Diophanative with exponents

Distributing on a circular table

Beautiful problem on midlines and isogonal

Game on coloring cells of a board.

10 numbers forming 500 &quot;good&quot; quadratic trinomials

Prove that (ab+c, bc+a, ca+b)=(a,b,c)

On a circumscribed pentagon

10 numbers forming 500 "good" quadratic trinomials