MathDB

1

lattice points on [O,N]

[0, 2002]

its ends and the point with coordinate

d

are marked, where

d

is a coprime number to

1001

. It is allowed to mark the midpoint of any segment with ends at the marked points, if its coordinate is integer. Is it possible, by repeating this operation several times, to mark all the integer points on a segment?
(10.7) On the segment

[0, 2002]

its ends and

n-1 > 0

integer points are marked so that the lengths of the segments into which the segment

[0, 2002]

is divided are corpime in the total (i.e., have no common divisor greater than

1

). It is allowed to divide any segment with marked ends into

n

equal parts and mark the division points if they are all integers. (The point can be marked a second time, but it remains marked.) Is it possible, by repeating this operation several times, mark all the integer points on the segment?
(11.8) On the segment

[0,N]

its ends and

2

more points are marked so that the lengths segments into which the segment

[0,N]

is divided are integer and coprime in total. If there are two marked points

A

and

B

such that the distance between them is a multiple of

3

, then we can divide from cutting

AB

by

3

equal parts, mark one of the division points and erase one of the points

A, B

. Is it true that for several such actions you can mark any predetermined integer point of the segment

[0,N]

?

11.6

1

n points, vectors, 2 player game - All-Russian MO 2002 Regional (R4) 11.6

There are

n > 1

points on the plane. Two take turns connecting more an unconnected pair of points by a vector of one of two possible directions. If after the next move of a player the sum of all drawn vectors is zero, then the second one wins; if it's another move is impossible, and there was no zero sum, then the first one wins. Who wins when played correctly?

11.5

1

no of roots of P(P(x)) = 0 vs P(x) = 0 - All-Russian MO 2002 Regional (R4) 11.5

Let

P(x)

be a polynomial of odd degree. Prove that the equation

P(P(x)) = 0

has at least as many different real roots as the equation

P(x) = 0

[hide=original wording]Пусть P(x) — многочлен нечетной степени. Докажите, что уравнение P(P(x)) = 0 имеет не меньше различных действительных корней, чем уравнение P(x) = 0

11.4

1

exists a column colored in n colors - All-Russian MO 2002 Regional (R4) 11.4

Each cell of the checkered plane is colored in one of

n^2

colors so that in any square of

n \times n

cells all colors occur. It is known that in some line all the colors occur. Prove that there exists a column colored in exactly

n

colors.

11.2

1

concurrency in a quadrangular pyramid , 5 points concyclic

The altitude of a quadrangular pyramid

SABCD

passes through the intersection point of the diagonals of its base

ABCD

. From the tops of the base perpendiculars

AA_1

,

BB_1

,

CC_1

,

DD_1

are dropped onto lines

SC

,

SD,

SA

and

SB

respectively. It turned out that the points

S

,

A_1

,

B_1

,

C_1

,

D_1

are different and lie on the same sphere. Prove that lines

AA_1

,

BB_1

,

CC_1

,

DD_1

pass through one point.

11.1

1

x^p + y^q is rational, for odd prime p All-Russian MO 2002 Regional (R4) 11.1

The real numbers

x

and

y

are such that for any distinct odd primes

p

and

q

the number

x^p + y^q

is rational. Prove that

x

and

y

are rational numbers.

10.5

1

n parabolas cut Ox - All-Russian MO 2002 Regional (R4) 10.5

Various points

x_1,..., x_n

(

n \ge 3

) are randomly located on the

Ox

axis. Construct all parabolas defined by the monic square trinomials and intersecting the Ox axis at these points (and not intersecting axis at other points). Let

y = f_1

,

...

,

y = f_m

are functions that define these parabolas. Prove that the parabola

y = f_1 +...+ f_m

intersects the

Ox

axis at two points.

10.4

1

sum a_k^3 <= (\sum a _k )^2

(10.4) A set of numbers

a_0, a_1,..., a_n

satisfies the conditions:

a_0 = 0

,

0 \le a_{k+1}- a_k \le 1

for

k = 0, 1, .. , n -1

. Prove the inequality

\sum_{k=1}^n a^3_k \le \left(\sum_{k=1}^n a_k \right)^2

(11.3) A set of numbers

a_0, a_1,..., a_n

satisfies the conditions:

a_0 = 0

,

a_{k+1} \ge a_k + 1

for

k = 0, 1, .. , n -1

. Prove the inequality

\sum_{k=1}^n a^3_k \ge \left(\sum_{k=1}^n a_k \right)^2

10.2

1

convex polygon that contains >=m^2+1 lattice pointa has m+1 collinear lattice

A convex polygon on a plane contains at least

m^2+1

points with integer coordinates. Prove that it contains

m+1

points with integer coordinates that lie on the same line.

10.3

1

line of 2 incenters perpendicular to angle bisector

The perpendicular bisector to side

AC

of triangle

ABC

intersects side

BC

at point

M

(see fig.). The bisector of angle

\angle AMB

intersects the circumcircle of triangle

ABC

at point

K

. Prove that the line passing through the centers of the inscribed circles triangles

AKM

and

BKM

, perpendicular to the bisector of angle

\angle AKB

. https://cdn.artofproblemsolving.com/attachments/b/4/b53ec7df0643a90b835f142d99c417a2a1dd45.png

9.6

1

segments bisecting areas of trapezoid, symmetric intersection points

Let

A'

be a point on one of the sides of the trapezoid

ABCD

such that line

AA'

divides the area of the trapezoid in half. Points

B'

,

C'

,

D'

ABCD

and

A'B'C'D'

are symmetrical wrt the midpoint of midline of trapezoid

ABCD

.

9.5

1

rearrange numbers 1 - 60 - All-Russian MO 2002 Regional (R4) 9.5

Is it possible to arrange the numbers

1, 2, . . . , 60

in that order, so that the sum of any two numbers between which there is one number, divisible by

2

, the sum of any two numbers between which there are two numbers divisible by

3

, . . . , the sum of any two numbers between which there is are there six numbers, divisible by

7

?

9.4

1

1 rectangle intersects all rectangles - All-Russian MO 2002 Regional (R4) 9.4

Located on the plane

\left[ \frac43 n \right]

rectangles with sides parallel to the coordinate axes. It is known that any rectangle intersects at least n rectangles. Prove that exists a rectangle that intersects all rectangles.

9.3

1

O,K,C,L, concyclic wanted, <CLO = < BLM starting with isosceles ABC

In an isosceles triangle

ABC

(

AB = BC

), point

O

is the center of the circumcircle. Point

M

lies on the segment

BO

, point

M'

is symmetric to

M

wrt the midpoint of

AB

. Point K is the intersection point of of

M'O

and

AB

. Point

L

lies on side BC such that

\angle CLO = \angle BLM

. Prove that points

O, K,B,L

lie on the same circle

8.8

1

100 and 99 g, weights - All-Russian MO 2002 Regional (R4) 8.8

Among

18

parts placed in a row, some three in a row weigh

99

g each, and all the rest weigh

100

g each. On a scale with an arrow, identify all

99

-gram parts.

8.7

1

Moskvich and Zaporozhets - All-Russian MO 2002 Regional (R4) 8.7

''Moskvich'' and ''Zaporozhets'' drove past the observer on the highway and the Niva moving towards them. It is known that when the Moskvich caught up with the observer, it was equidistant from the Zaporozhets and the Niva, and when the Niva caught up with the observer, it was equal. but removed from ''Moskvich'' and ''Zaporozhets''. Prove that ''Zaporozhets'' at the moment of passing by the observer was equidistant from the Niva and ''Moskvich''.

8.6

1

8 points at extensions are concyclic - All-Russian MO 2002 Regional (R4) 8.6

Each side of the convex quadrilateral was continued into both sides and on all eight extensions set aside equal segments. It turned out that the resulting

8

points are the outer ends of the construction the given segments are different and lie on the same circle. Prove that the original quadrilateral is a square.

8.5

1

1234 -> 2002 - All-Russian MO 2002 Regional (R4) 8.5

The four-digit number written on the board can be replaced by another, adding one to its two adjacent digits, if neither of these digits is not equal to

9

; or, subtracting one from the adjacent two digits, if none of them is equal to

0

. Is it possible using such operations from does the number

1234

get the number

2002

?

8.4

1

regular triangles on side of triangle - All-Russian MO 2002 Regional (R4) 8.4

Given a triangle

ABC

with pairwise distinct sides. on his on the sides, regular triangles

ABC_1

,

BCA_1

,

CAB_1

. are constructed externally. Prove that triangle

A_1B_1C_1

cannot be regular.

8.3

1

coins and 11 empty boxes - All-Russian MO 2002 Regional (R4) 8.3

There are

11

empty boxes. In one move you can put one coin in some 10 of them. Two people play and take turns. Wins the one after which for the first time there will be

21

coins in one of the boxes. Who wins when played correctly?

8.1

1

numbers in 9 x 2002 - All-Russian MO 2002 Regional (R4) 8.1

Is it possible to fill all the cells of the table

9 \times 2002

with natural numbers so that the sum of the numbers in any column and the sum of the numbers in any string would be prime numbers?

11.7

1

Pair of Tangent Circles!

Given a convex quadrilateral

ABCD

.Let

\ell_A,\ell_B,\ell_C,\ell_D

be exterior angle bisectors of quadrilateral

ABCD

. Let

\ell_A \cap \ell_B=K,\ell_B \cap \ell_C=L,\ell_C \cap \ell_D=M,\ell_D \cap \ell_A=N

.Prove that if circumcircles of triangles

ABK

and

CDM

be externally tangent to each other then circumcircles of the triangles

BCL

and

DAN

are externally tangent to each other.(L.Emelyanov)

10.8

1

coloring 10*10 square board

what maximal number of colors can we use in order to color squares of 10 *10 square board so that each row or column contains squares of at most 5 different colors?

9.2

1

All-Russian Olympiads 2002 p2

A monic quadratic polynomial

f

with integer coefficients attains prime values at three consecutive integer points.show that it attains a prime value at some other integer point as well.

8.2

1

All-Russian Olympiads 2002 p1

each cells in a

9\times 9

grid is painted either blue or red.two cells are called diagonal neighbors if their intersection is exactly a point.show that some cell has exactly two red neighbors,or exactly two blue neighbors, or both.

10.1

1

Russia 2002

What is the largest possible length of an arithmetic progression of positive integers

a_{1}, a_{2},\cdots , a_{n}

with difference

2

, such that {a_{k}}^{2}\plus{}1 is prime for k \equal{} 1, 2, . . . , n?

2002 All-Russian Olympiad Regional Round

Subcontests

lattice points on [O,N]

n points, vectors, 2 player game - All-Russian MO 2002 Regional (R4) 11.6

no of roots of P(P(x)) = 0 vs P(x) = 0 - All-Russian MO 2002 Regional (R4) 11.5

exists a column colored in n colors - All-Russian MO 2002 Regional (R4) 11.4

concurrency in a quadrangular pyramid , 5 points concyclic

x^p + y^q is rational, for odd prime p All-Russian MO 2002 Regional (R4) 11.1

n parabolas cut Ox - All-Russian MO 2002 Regional (R4) 10.5

sum a_k^3 <= (\sum a _k )^2

convex polygon that contains >=m^2+1 lattice pointa has m+1 collinear lattice

line of 2 incenters perpendicular to angle bisector

segments bisecting areas of trapezoid, symmetric intersection points

rearrange numbers 1 - 60 - All-Russian MO 2002 Regional (R4) 9.5

1 rectangle intersects all rectangles - All-Russian MO 2002 Regional (R4) 9.4

O,K,C,L, concyclic wanted, <CLO = < BLM starting with isosceles ABC

100 and 99 g, weights - All-Russian MO 2002 Regional (R4) 8.8

Moskvich and Zaporozhets - All-Russian MO 2002 Regional (R4) 8.7

8 points at extensions are concyclic - All-Russian MO 2002 Regional (R4) 8.6

1234 -> 2002 - All-Russian MO 2002 Regional (R4) 8.5

regular triangles on side of triangle - All-Russian MO 2002 Regional (R4) 8.4

coins and 11 empty boxes - All-Russian MO 2002 Regional (R4) 8.3

numbers in 9 x 2002 - All-Russian MO 2002 Regional (R4) 8.1

Pair of Tangent Circles!

coloring 10*10 square board

All-Russian Olympiads 2002 p2

All-Russian Olympiads 2002 p1

Russia 2002

2002 All-Russian Olympiad Regional Round

Subcontests

lattice points on [O,N]

n points, vectors, 2 player game - All-Russian MO 2002 Regional (R4) 11.6

no of roots of P(P(x)) = 0 vs P(x) = 0 - All-Russian MO 2002 Regional (R4) 11.5

exists a column colored in n colors - All-Russian MO 2002 Regional (R4) 11.4

concurrency in a quadrangular pyramid , 5 points concyclic

x^p + y^q is rational, for odd prime p All-Russian MO 2002 Regional (R4) 11.1

n parabolas cut Ox - All-Russian MO 2002 Regional (R4) 10.5

sum a_k^3 &lt;= (\sum a _k )^2

convex polygon that contains &gt;=m^2+1 lattice pointa has m+1 collinear lattice

line of 2 incenters perpendicular to angle bisector

segments bisecting areas of trapezoid, symmetric intersection points

rearrange numbers 1 - 60 - All-Russian MO 2002 Regional (R4) 9.5

1 rectangle intersects all rectangles - All-Russian MO 2002 Regional (R4) 9.4

O,K,C,L, concyclic wanted, &lt;CLO = &lt; BLM starting with isosceles ABC

100 and 99 g, weights - All-Russian MO 2002 Regional (R4) 8.8

Moskvich and Zaporozhets - All-Russian MO 2002 Regional (R4) 8.7

8 points at extensions are concyclic - All-Russian MO 2002 Regional (R4) 8.6

1234 -&gt; 2002 - All-Russian MO 2002 Regional (R4) 8.5

regular triangles on side of triangle - All-Russian MO 2002 Regional (R4) 8.4

coins and 11 empty boxes - All-Russian MO 2002 Regional (R4) 8.3

numbers in 9 x 2002 - All-Russian MO 2002 Regional (R4) 8.1

Pair of Tangent Circles!

coloring 10*10 square board

All-Russian Olympiads 2002 p2

All-Russian Olympiads 2002 p1

Russia 2002

sum a_k^3 <= (\sum a _k )^2

convex polygon that contains >=m^2+1 lattice pointa has m+1 collinear lattice

O,K,C,L, concyclic wanted, <CLO = < BLM starting with isosceles ABC

1234 -> 2002 - All-Russian MO 2002 Regional (R4) 8.5