MathDB

1

|XA|, |XB| >=0.999|AB| - All-Russian MO 2001 Regional (R4) 11.7

S

on the plane, and any

1\times 1

square contains a finite number of points from the set

S

. Prove that there are two different points

A

and

B

from

S

such that for any other point

X

from

S

the following inequalities hold:

|XA|, |XB| \ge 0.999|AB|.

11.8

1

set of 177 3-digit - All-Russian MO 2001 Regional (R4) 11.8

Prove that in any set consisting of

117

pairwise distinct three-digit numbers, you can choose

4

pairwise disjoint subsets in which the sums of numbers are equal.

11.6

1

concurrency in a tetrahedron,

Prove that if two segments of a tetrahedron, going from the ends of some edge to the centers of the inscribed circles of opposite faces, intersect, then the segments issued from the ends of the crossing with it edges to the centers of the inscribed circles of the other two faces, also intersect.

11.5

1

x_{n+1} = n sin x_n+ 1 non periodic - All-Russian MO 2001 Regional (R4) 11.5

Given a sequence

\{x_k\}

such that

x_1 = 1

,

x_{n+1} = n \sin x_n+ 1

. Prove that the sequence is non-periodic.

11.2

1

3 roots for f(f(x)) = 0, 7 f(f(x)) = 0 - All-Russian MO 2001 Regional (R4) 11.2

The monic quadratic trinomial

f(x)

has

2

different roots. Could it be that the equation

f(f(x)) = 0

has

3

different root, and the equation

f(f(f(x))) = 0

has

7

different roots?

11.1

1

p + q = (p -q)^3 - All-Russian MO 2001 Regional (R4) 11.1

Find all prime numbers

p

and

q

such that

p + q = (p -q)^3.

10.8

1

numbers on 1000 arcs - All-Russian MO 2001 Regional (R4) 10.8

There are a thousand non-intersecting arcs on a circle, and on each of them contains two natural numbers. Sum of numbers of each arc is divided by the product of the numbers of the arc following it clockwise arrow. What is the largest possible value of the largest number written?

10.7

1

rook-connected set of 100 cells - All-Russian MO 2001 Regional (R4) 10.7

We call a set of cells on a checkered plane rook-connected if from any of its cells one can get to any other by moving along the cells of this set by moving the rook (the rook is allowed to fly through fields that do not belong to our set). Prove that a rook-connected set of

100

cells can be divided into pairs of cells, lying in one row or in one column.

10.5

1

ax^2+bx+c, (c-b)x^2 + (c- a)x +(a + b) - All-Russian MO 2001 Regional (R4) 10.5

Given integers

a

,

b

and

c

,

c\ne b

. It is known that the square trinomials

ax^2 + bx + c

and

(c-b)x^2 + (c- a)x + (a + b)

have a common root (not necessarily integer). Prove that

a+b+2c

is divisible by

3

.

10.4

1

10 lines, 3 families All-Russian MO 2001 Regional (R4) 10.4

Three families of parallel lines are drawn,

10

lines each, are drawn. What is the greatest number of triangles they can cut from plane?

10.3

1

3 colors for naturals, 2000(a + b) = c - All-Russian MO 2001 Regional (R4) 10.3

Describe all the ways to color each natural number as one of three colors so that the following condition is satisfied: if the numbers

a

,

b

and

c

(not necessarily different) satisfy the condition

2000(a + b) = c

, then they either all the same color or three different colors

10.1

1

f(A)=f(B), sum of lengths of polygons - All-Russian MO 2001 Regional (R4) 10.1

The lengths of the sides of the polygon are

a_1

,

a_2

,.

..

,

a_n

. The square trinomial

f(x)

is such that

f(a_1) = f(a_2 +...+ a_n)

. Prove that if

A

is the sum of the lengths of several sides of a polygon,

B

is the sum of the lengths of its remaining sides, then

f(A) = f(B)

.

9.2

1

f = x^2 + ax + b - All-Russian MO 2001 Regional (R4) 9.2

Petya and Kolya play the following game: they take turns changing one of the coefficients

a

or

b

of the quadratic trinomial

f = x^2 + ax + b

: Petya is on

1

, Kolya is on

1

or

3

. Kolya wins if after the move of one of the players a trinomial is obtained that has whole roots. Is it true that Kolya can win for any initial integer odds

a

and

b

regardless of Petya's game?
[hide=original wording]Петя и Коля играют в следующую игру: они по очереди изменяют один из коэффициентов a или b квадратного трехчлена f = x^2 + ax + b: Петя на 1, Коля- на 1 или на 3. Коля выигрывает, если после хода одного из игроков получается трехчлен, имеющий целые корни. Верно ли, что Коля может выигратьпр и любых начальных целых коэффициентах a и b независимо от игры Пети?

9.8

1

<=100 perfect squares - All-Russian MO 2001 Regional (R4) 9.8

Sasha wrote a non-zero number on the board and added it to it on the right, one non-zero digit at a time, until he writes out a million digits. Prove that an exact square has been written on the board no more than

100

times.

9.6

1

product of divisors ends in 2001 0s - All-Russian MO 2001 Regional (R4) 9.6

Is there such a natural number that the product of all its natural divisors (including

1

and the number itself) ends exactly in

2001

zeros?

9.5

1

2 points in a convex pentagon - All-Russian MO 2001 Regional (R4) 9.5

Two points are selected in a convex pentagon. Prove that you can choose a quadrilateral with vertices at the vertices of a pentagon so that both selected points fall into it.

9.4

1

target of 100 equal equilaterals - All-Russian MO 2001 Regional (R4) 9.4

The target is a triangle divided by three families of parallel lines into

100

equal regular triangles with single sides. A sniper shoots at a target. He aims at triangle and hits either it or one of the sides adjacent to it. He sees the results of his shooting and can choose when stop shooting. What is the greatest number of triangles he can with a guarantee of hitting five times?

11.3

1

common tangent of 3 circles - All-Russian MO 2001 Regional (R4) 11.3

Let

AD

be the angle bisector of triangle

ABC

, and let the line

\ell

touch circumcircles of triangles

ADB

and

ADC

at points

M

and

N

accordingly. Prove that the circle passing through the midpoints of the segments

BD

,

DC

and

MN

is tangent to the line

\ell

.

10.6

1

tangent at Q is // to AC - All-Russian MO 2001 Regional (R4) 10.6

Given triangle

ABC

. Point

B_1

is marked on line

AC

so that

AB = AB_1

, while

B_1

and

C

are on the same side of

A

. Through points

C

,

B_1

and the foot of the bisector of angle

A

of triangle

ABC

, a circle

\omega

is drawn, intersecting for second time the circle circumscribed around triangle

ABC

, at point

Q

. Prove that the tangent drawn to

\omega

at point

Q

is parallel to

AC

.

10.2

1

// lines connecting 2 centers, # - All-Russian MO 2001 Regional (R4) 10.2

In parallelogram

ABCD

, point

K

is marked on diagonal

AC

. Circle

s_1

passes through point

K

and touches lines

AB

and

AD

(

s_1

intersects the diagonal

AC

for the second time on the segment

AK

). Circle

s_2

passes through point

K

and touches lines

CB

and

CD

(

s_2

intersects for the second time diagonal

AC

on segment

KC

). Prove that for all positions of the point

K

on the diagonal

AC

, the straight lines connecting the centers of circles

s_1

and

s_2

, will be parallel to each other.

9.7

1

OM tangent to circle wanted - All-Russian MO 2001 Regional (R4) 9.7

A circle inscribed in an angle with vertex

O

touches its sides at points

A

and

B

,

K

is an arbitrary point on the smaller of the two arcs

AB

of this circle. On the line

OB

a point

L

is taken such that the lines

OA

and

KL

are parallel. Let

M

be the intersection point of the circle

\omega

circumscribed around triangle

KLB

, with line

AK

, with

M

different from

K

. Prove that line

OM

touches circle

\omega

.

9.3

1

DQ bisects <CDA of ABCD#, AM = NC - All-Russian MO 2001 Regional (R4) 9.3

In parallelogram

ABCD

, points

M

and

N

are selected on sides

AB

and

BC

respectively so that

AM = NC

,

Q

is the intersection point of segments

AN

and

CM

. Prove that

DQ

is the bisector of angle

D

.

8.4

1

removing rectangles from mxn - All-Russian MO 2001 Regional (R4) 8.4

An angle of size

n \times m

, where

m, n \ge 2

, is called a figure, resulting from a rectangle of size

n \times m

cells by removing the rectangle size

(n - 1) \times (m - 1)

cells. Two players take turns making moves consisting in painting in a corner an arbitrary non-zero number of cells forming a rectangle or square.

8.8

1

isosceles by 3 parts of any triangle - All-Russian MO 2001 Regional (R4) 8.8

Prove that any triangle can be cut by at most into

3

parts, from which an isosceles triangle is formed.

8.7

1

4 colors in 5x5 board - All-Russian MO 2001 Regional (R4) 8.7

Is it possible to paint the cells of a

5\times 5

board in

4

colors so that the cells standing at the intersection of any two rows and any two columns were painted in at least

3

colors?

8.6

1

sum of digits | n, n+1, n+2, n+3 - All-Russian MO 2001 Regional (R4) 8.6

We call a natural number

n

good if each of the numbers

n

,

n+1

,

n+2

and

n+3

are divided by the sum of their digits. (For example,

n = 60398

is good.) Does the penultimate digit of a good number ending in eight have to be nine?

8.5

1

|ax+b|+|cx+d|=|ex+f| - All-Russian MO 2001 Regional (R4) 8.5

Let

a, b, c, d, e

and

f

be some numbers, and

a \cdot c \cdot e \ne 0

.It is known that the values of the expressions

|ax+b|+|cx+d|

and

|ex+f|

equal at all values of

x

. Prove that

ad = bc

.

8.3

1

min max angle convex pentagon - All-Russian MO 2001 Regional (R4) 8.3

All sides of a convex pentagon are equal, and all angles are different. Prove that the maximum and minimum angles are adjacent to the same side of the pentagon.

8.2

1

n numbers 1s and 2s in a circle - All-Russian MO 2001 Regional (R4) 8.2

N

numbers - ones and twos - are arranged in a circle. We mean a number formed by several digits arranged in a row (clockwise or counterclockwise). For what is the smallest value of

N

, all four-digit numbers whose writing contains only numbers

1

and

2

, could they be among those shown?

8.1

1

nubmers 1-10 in a row - All-Russian MO 2001 Regional (R4) 8.1

Is it possible to have numbers

1, 2,..., 10

put in a row in some order so that each of them, starting from the second, differs from the previous one by a whole number of percent?

2001 All-Russian Olympiad Regional Round

Subcontests

|XA|, |XB| >=0.999|AB| - All-Russian MO 2001 Regional (R4) 11.7

set of 177 3-digit - All-Russian MO 2001 Regional (R4) 11.8

concurrency in a tetrahedron,

x_{n+1} = n sin x_n+ 1 non periodic - All-Russian MO 2001 Regional (R4) 11.5

3 roots for f(f(x)) = 0, 7 f(f(x)) = 0 - All-Russian MO 2001 Regional (R4) 11.2

p + q = (p -q)^3 - All-Russian MO 2001 Regional (R4) 11.1

numbers on 1000 arcs - All-Russian MO 2001 Regional (R4) 10.8

rook-connected set of 100 cells - All-Russian MO 2001 Regional (R4) 10.7

ax^2+bx+c, (c-b)x^2 + (c- a)x +(a + b) - All-Russian MO 2001 Regional (R4) 10.5

10 lines, 3 families All-Russian MO 2001 Regional (R4) 10.4

3 colors for naturals, 2000(a + b) = c - All-Russian MO 2001 Regional (R4) 10.3

f(A)=f(B), sum of lengths of polygons - All-Russian MO 2001 Regional (R4) 10.1

f = x^2 + ax + b - All-Russian MO 2001 Regional (R4) 9.2

<=100 perfect squares - All-Russian MO 2001 Regional (R4) 9.8

product of divisors ends in 2001 0s - All-Russian MO 2001 Regional (R4) 9.6

2 points in a convex pentagon - All-Russian MO 2001 Regional (R4) 9.5

target of 100 equal equilaterals - All-Russian MO 2001 Regional (R4) 9.4

common tangent of 3 circles - All-Russian MO 2001 Regional (R4) 11.3

tangent at Q is // to AC - All-Russian MO 2001 Regional (R4) 10.6

// lines connecting 2 centers, # - All-Russian MO 2001 Regional (R4) 10.2

OM tangent to circle wanted - All-Russian MO 2001 Regional (R4) 9.7

DQ bisects <CDA of ABCD#, AM = NC - All-Russian MO 2001 Regional (R4) 9.3

removing rectangles from mxn - All-Russian MO 2001 Regional (R4) 8.4

isosceles by 3 parts of any triangle - All-Russian MO 2001 Regional (R4) 8.8

4 colors in 5x5 board - All-Russian MO 2001 Regional (R4) 8.7

sum of digits | n, n+1, n+2, n+3 - All-Russian MO 2001 Regional (R4) 8.6

|ax+b|+|cx+d|=|ex+f| - All-Russian MO 2001 Regional (R4) 8.5

min max angle convex pentagon - All-Russian MO 2001 Regional (R4) 8.3

n numbers 1s and 2s in a circle - All-Russian MO 2001 Regional (R4) 8.2

nubmers 1-10 in a row - All-Russian MO 2001 Regional (R4) 8.1

2001 All-Russian Olympiad Regional Round

Subcontests

|XA|, |XB| &gt;=0.999|AB| - All-Russian MO 2001 Regional (R4) 11.7

set of 177 3-digit - All-Russian MO 2001 Regional (R4) 11.8

concurrency in a tetrahedron,

x_{n+1} = n sin x_n+ 1 non periodic - All-Russian MO 2001 Regional (R4) 11.5

3 roots for f(f(x)) = 0, 7 f(f(x)) = 0 - All-Russian MO 2001 Regional (R4) 11.2

p + q = (p -q)^3 - All-Russian MO 2001 Regional (R4) 11.1

numbers on 1000 arcs - All-Russian MO 2001 Regional (R4) 10.8

rook-connected set of 100 cells - All-Russian MO 2001 Regional (R4) 10.7

ax^2+bx+c, (c-b)x^2 + (c- a)x +(a + b) - All-Russian MO 2001 Regional (R4) 10.5

10 lines, 3 families All-Russian MO 2001 Regional (R4) 10.4

3 colors for naturals, 2000(a + b) = c - All-Russian MO 2001 Regional (R4) 10.3

f(A)=f(B), sum of lengths of polygons - All-Russian MO 2001 Regional (R4) 10.1

f = x^2 + ax + b - All-Russian MO 2001 Regional (R4) 9.2

&lt;=100 perfect squares - All-Russian MO 2001 Regional (R4) 9.8

product of divisors ends in 2001 0s - All-Russian MO 2001 Regional (R4) 9.6

2 points in a convex pentagon - All-Russian MO 2001 Regional (R4) 9.5

target of 100 equal equilaterals - All-Russian MO 2001 Regional (R4) 9.4

common tangent of 3 circles - All-Russian MO 2001 Regional (R4) 11.3

tangent at Q is // to AC - All-Russian MO 2001 Regional (R4) 10.6

// lines connecting 2 centers, # - All-Russian MO 2001 Regional (R4) 10.2

OM tangent to circle wanted - All-Russian MO 2001 Regional (R4) 9.7

DQ bisects &lt;CDA of ABCD#, AM = NC - All-Russian MO 2001 Regional (R4) 9.3

removing rectangles from mxn - All-Russian MO 2001 Regional (R4) 8.4

isosceles by 3 parts of any triangle - All-Russian MO 2001 Regional (R4) 8.8

4 colors in 5x5 board - All-Russian MO 2001 Regional (R4) 8.7

sum of digits | n, n+1, n+2, n+3 - All-Russian MO 2001 Regional (R4) 8.6

|ax+b|+|cx+d|=|ex+f| - All-Russian MO 2001 Regional (R4) 8.5

min max angle convex pentagon - All-Russian MO 2001 Regional (R4) 8.3

n numbers 1s and 2s in a circle - All-Russian MO 2001 Regional (R4) 8.2

nubmers 1-10 in a row - All-Russian MO 2001 Regional (R4) 8.1

|XA|, |XB| >=0.999|AB| - All-Russian MO 2001 Regional (R4) 11.7

<=100 perfect squares - All-Russian MO 2001 Regional (R4) 9.8

DQ bisects <CDA of ABCD#, AM = NC - All-Russian MO 2001 Regional (R4) 9.3