MathDB

1

|x+a|+|x+y+b|+|y+c|>|x|+|x+y|+|y| - All-Russian MO 1999 Regional (R4) 11.5

a, b

and

c

such that for all real

x

and

y

the following inequality holds:

|x + a| + |x + y + b| + |y + c| > |x| + |x + y| + |y|?

11.8

1

>=2 integer points for polynomial - All-Russian MO 1999 Regional (R4) 11.8

For some polynomial there is an infinite set its values, each of which takes at least at two integer points. Prove that there is at most one the integer value that a polynomial takes at exactly one integer point.

11.6

1

4 colors in 50x50 square - All-Russian MO 1999 Regional (R4) 11.6

The cells of a

50\times 50

square are painted in four colors. Prove that there is a cell, on four sides of which (i.e. top, bottom, left and on the right) there are cells of the same color as it (not necessarily adjacent to this cell).

11.4

1

circumscribed polyhedron around a sphere has <= 6 large faces

A polyhedron is circumscribed around a sphere. Let's call its face large if the projection of the sphere onto the plane of the face falls entirely within the face. Prove that there are no more than 6 large faces.

11.3

1

talker is friend with silent - All-Russian MO 1999 Regional (R4) 11.3

In the class, every talker is friends with at least one silent person. At this chatterbox is silent if there is an odd number of his friends in the office —silent. Prove that the teacher can invite you to an elective class without less than half the class so that all talkers are silent.
[hide=original wording]В классе каждый болтун дружит хотя бы с одним молчуном. При этом болтун молчит, если в кабинете находится нечетное число его друзей - молчунов. Докажите, что учительмо жет пригласитьна факультатив не менее половины класса так, чтобы все болтуны молчали

11.1

1

f(x)+f(ax) is continuous - All-Russian MO 1999 Regional (R4) 11.1

The function

f(x)

, defined on the entire real line, is known but that for any

a > 1

the function

f(x)+f(ax)

is continuous on the entire line. Prove that

f(x)

is also continuous along the entire line.

10.6

1

CD = CB_1, touchpoints of incircle - All-Russian MO 1999 Regional (R4) 10.6

Triangle

ABC

has an inscribed circle tangent to sides

AB

,

AC

and

BC

at points

C_1

,

B_1

and

A_1

respectively. Let

K

be a point on the circle diametrically opposite to point

C_1

,

D

be the intersection point of lines

B_1C_1

and

A_1K

. Prove that

CD = CB_1

.

10.2

1

collinear circumcenters - All-Russian MO 1999 Regional (R4) 10.2

Given a circle

\omega

, a point

A

lying inside

\omega

, and point

B

(

B \ne A

). All possible triangles

BXY

are considered, such that the points

X

and

Y

lie on

\omega

and the chord

XY

passes through the point

A

. Prove that the centers of the circumcircles of the triangles

BXY

lie on the same straight line.

10.8

1

marked number in segment 1999 cm - All-Russian MO 1999 Regional (R4) 10.8

Some natural numbers are marked. It is known that on every a segment of the number line of length

1999

has a marked number. Prove that there is a pair of marked numbers, one of which is divisible by the other.

10.7

1

n names of candidates, elections - All-Russian MO 1999 Regional (R4) 10.7

Each voter in an election puts

n

names of candidates on the ballot. There are

n + 1

at the polling station urn. After the elections it turned out that each ballot box contained at least at least one ballot, for every choice of the

(n + 1)

-th ballot, one from each ballot box, there is a candidate whose surname appears in each of the selected ballots. Prove that in at least one ballot box all ballots contain the name of the same candidate.

10.5

1

9 perfect squares - - All-Russian MO 1999 Regional (R4) 10.5

Are there

10

different integers such that all the sums made up of

9

of them are perfect squares?

10.3

1

n points in space, 3d combo geo - All-Russian MO 1999 Regional (R4) 10.3

There are

n

points in general position in space (no three lie on the same straight line, no four lie in the same plane). A plane is drawn through every three of them. Prove that If you take any whatever

n-3

points in space, there is a plane from those drawn that does not contain any of these

n - 3

points.

9.5

1

5 colors for checkered plane - All-Russian MO 1999 Regional (R4) 9.5

All cells of the checkered plane are painted in

5

colors so that in any figure of the species https://cdn.artofproblemsolving.com/attachments/f/f/49b8d6db20a7e9cca7420e4b51112656e37e81.png all colors are different. Prove that in any figure of the species \begin{tabular}{ | l | c| c | c | r| } \hline & & & &\\ \hline \end{tabular}, all colors are different..

9.3

1

sum 1/x^k >= sum x^k - All-Russian MO 1999 Regional (R4) 9.3

The product of positive numbers

x, y

and

z

is equal to

1

. Prove that if it holds that

\frac1x +\frac1y + \frac1z \ge x + y + z,

then for any natural

k

, holds the inequality

\frac{1}{x^k} +\frac{1}{y^k} + \frac{1}{z^k} \ge x^k + y^k + z^k.

9.7

1

n=a-b, no of prime factors - All-Russian MO 1999 Regional (R4) 9.7

Prove that every natural number is the difference of two natural numbers that have the same number of prime factors. (Each prime divisor is counted once, for example, the number

12

has two prime factors:

2

and

3

.)

9.4

1

8x8 maze - All-Russian MO 1999 Regional (R4) 9.4

The maze is an

8 \times 8

square, each cell contains

1 \times 1

which has one of four arrows drawn (up, down, right, left). The upper side of the upper right cell is the exit from the maze.In the lower left cell there is a chip that, with each move, moves one square in the direction indicated by the arrow. After each move, the shooter in the cell in which there was just a chip rotates

90^o

clockwise. If a chip must move, taking it outside the

8 \times 8

square, it remains in place, and the arrow also rotates

90^o

clockwise. Prove that sooner or later, the chip will come out of the maze.

9.8

1

QO_|_AC if QP _|_ KM, QM//BO - All-Russian MO 1999 Regional (R4) 9.8

In triangle

ABC

(

AB > BC

),

K

and

M

are the midpoints of sides

AB

and

AC

,

O

is the point of intersection of the angle bisectors. Let

P

be the intersection point of lines

KM

and

CO

, and the point

Q

is such that

QP \perp KM

and

QM \parallel BO

. Prove that

QO \perp AC

.

9.2

1

B, E, D, F concyclic if AB = AD, BE = EC - All-Russian MO 1999 Regional (R4) 9.2

In triangle

ABC

, on side

AC

there are points

D

and

E

, that

AB = AD

and

BE = EC

(

E

between

A

and

D

). Point

F

is midpoint of arc

BC

of circumcircle of triangle

ABC

. Prove that the points

B, E, D, F

lie on the same circle.

9.1

1

n numbers in a circle, digits - All-Russian MO 1999 Regional (R4) 9.1

All natural numbers from

1

to

N

,

N \ge 2

are written out in a certain order in a circle. Moreover, for any pair of neighboring numbers there is at least one digit appearing in the decimal notation of each of them. Find the smallest possible value of

N

.

8.8

1

54 squares to cover 3x3x3 - All-Russian MO 1999 Regional (R4) 8.8

An open chain was made from

54

identical single cardboard squares, connecting them hingedly at the vertices. Any square (except for the extreme ones) is connected to its neighbors by two opposite vertices. Is it possible to completely cover a

3\times 3 \times3

surface with this chain of squares?

8.7

1

2 player domino game - All-Russian MO 1999 Regional (R4) 8.7

The box contains a complete set of dominoes. Two players take turns choosing one dice from the box and placing them on the table, applying them to the already laid out chain on either of the two sides according to the rules of domino. The one who cannot make his next move loses. Who will win if they both played correctly?

8.5

1

cannot divide 1-15 in 2 groups - All-Russian MO 1999 Regional (R4) 8.5

Prove that the numbers from

1

to

15

cannot be divided into two groups:

A

of

2

numbers and

B

of

13

numbers such that the sum of the numbers in group

B

is equal to product of numbers in group

A

.

8.4

1

40 identical gas cylinders - All-Russian MO 1999 Regional (R4) 8.4

There are

40

identical gas cylinders, gas pressure values in which we are unknown and may be evil. It is allowed to connect any cylinders with each other in an amount not exceeding a given natural number

k

, and then separate them; while the pressure gas in the connected cylinders is set equal to the arithmetic average of the pressures in them before the connection. At what minimum

k

is there a way to equalize the pressures in all

40

cylinders, regardless of initial pressure distribution in the cylinders?

8.2

1

n=1+2+...+A - All-Russian MO 1999 Regional (R4) 8.2

The natural number

A

has three digits added to its right. The resulting number turned out to be equal to the sum of all natural numbers from

1

to

A

. Find

A

.

8.1

1

father and two sons visit grandmother - All-Russian MO 1999 Regional (R4) 8.`

A father and two sons went to visit their grandmother, who Raya lives

33

km from the city. My father has a motor roller, the speed of which

25

km/h, and with a passenger -

20

km/h (with two passengers on a scooter It’s impossible to move). Each of the brothers walks along the road at a speed of

5

km/h. Prove that all three can get to grandma's in

3

hours

8.6

1

<CA_1B=90^o, symmetric points wrt lines- All-Russian MO 1999 Regional (R4) 8.6

Given triangle

ABC

. Point

A_1

is symmetric to vertex

A

wrt line

BC

, and point

C_1

is symmetric to vertex

C

wrt line

AB

. Prove that if points

A_1

,

B

and

C_1

lie on the same line and

C_1B = 2A_1B

, then angle

\angle CA_1B

is right.

8.3

1

medians of inscribed triangle // sides - All-Russian MO 1999 Regional (R4) 8.3

On sides

BC

,

CA

,

AB

of triangle

ABC

, points

A_1

,

B_1

,

C_1

are chosen, respectively, so that the medians

A_1A_2

,

B_1B_2

,

C_1C_2

of the triangle

A_1B_1C_1

are respectively parallel to straight lines

AB

,

BC

,

CA

. Determine in what ratio points

A_1

,

B_1

,

C_1

divide the sides of the triangle

ABC

.

1999 All-Russian Olympiad Regional Round

Subcontests

|x+a|+|x+y+b|+|y+c|>|x|+|x+y|+|y| - All-Russian MO 1999 Regional (R4) 11.5

>=2 integer points for polynomial - All-Russian MO 1999 Regional (R4) 11.8

4 colors in 50x50 square - All-Russian MO 1999 Regional (R4) 11.6

circumscribed polyhedron around a sphere has <= 6 large faces

talker is friend with silent - All-Russian MO 1999 Regional (R4) 11.3

f(x)+f(ax) is continuous - All-Russian MO 1999 Regional (R4) 11.1

CD = CB_1, touchpoints of incircle - All-Russian MO 1999 Regional (R4) 10.6

collinear circumcenters - All-Russian MO 1999 Regional (R4) 10.2

marked number in segment 1999 cm - All-Russian MO 1999 Regional (R4) 10.8

n names of candidates, elections - All-Russian MO 1999 Regional (R4) 10.7

9 perfect squares - - All-Russian MO 1999 Regional (R4) 10.5

n points in space, 3d combo geo - All-Russian MO 1999 Regional (R4) 10.3

5 colors for checkered plane - All-Russian MO 1999 Regional (R4) 9.5

sum 1/x^k >= sum x^k - All-Russian MO 1999 Regional (R4) 9.3

n=a-b, no of prime factors - All-Russian MO 1999 Regional (R4) 9.7

8x8 maze - All-Russian MO 1999 Regional (R4) 9.4

QO_|_AC if QP _|_ KM, QM//BO - All-Russian MO 1999 Regional (R4) 9.8

B, E, D, F concyclic if AB = AD, BE = EC - All-Russian MO 1999 Regional (R4) 9.2

n numbers in a circle, digits - All-Russian MO 1999 Regional (R4) 9.1

54 squares to cover 3x3x3 - All-Russian MO 1999 Regional (R4) 8.8

2 player domino game - All-Russian MO 1999 Regional (R4) 8.7

cannot divide 1-15 in 2 groups - All-Russian MO 1999 Regional (R4) 8.5

40 identical gas cylinders - All-Russian MO 1999 Regional (R4) 8.4

n=1+2+...+A - All-Russian MO 1999 Regional (R4) 8.2

father and two sons visit grandmother - All-Russian MO 1999 Regional (R4) 8.`

<CA_1B=90^o, symmetric points wrt lines- All-Russian MO 1999 Regional (R4) 8.6

medians of inscribed triangle // sides - All-Russian MO 1999 Regional (R4) 8.3

1999 All-Russian Olympiad Regional Round

Subcontests

|x+a|+|x+y+b|+|y+c|&gt;|x|+|x+y|+|y| - All-Russian MO 1999 Regional (R4) 11.5

&gt;=2 integer points for polynomial - All-Russian MO 1999 Regional (R4) 11.8

4 colors in 50x50 square - All-Russian MO 1999 Regional (R4) 11.6

circumscribed polyhedron around a sphere has &lt;= 6 large faces

talker is friend with silent - All-Russian MO 1999 Regional (R4) 11.3

f(x)+f(ax) is continuous - All-Russian MO 1999 Regional (R4) 11.1

CD = CB_1, touchpoints of incircle - All-Russian MO 1999 Regional (R4) 10.6

collinear circumcenters - All-Russian MO 1999 Regional (R4) 10.2

marked number in segment 1999 cm - All-Russian MO 1999 Regional (R4) 10.8

n names of candidates, elections - All-Russian MO 1999 Regional (R4) 10.7

9 perfect squares - - All-Russian MO 1999 Regional (R4) 10.5

n points in space, 3d combo geo - All-Russian MO 1999 Regional (R4) 10.3

5 colors for checkered plane - All-Russian MO 1999 Regional (R4) 9.5

sum 1/x^k &gt;= sum x^k - All-Russian MO 1999 Regional (R4) 9.3

n=a-b, no of prime factors - All-Russian MO 1999 Regional (R4) 9.7

8x8 maze - All-Russian MO 1999 Regional (R4) 9.4

QO_|_AC if QP _|_ KM, QM//BO - All-Russian MO 1999 Regional (R4) 9.8

B, E, D, F concyclic if AB = AD, BE = EC - All-Russian MO 1999 Regional (R4) 9.2

n numbers in a circle, digits - All-Russian MO 1999 Regional (R4) 9.1

54 squares to cover 3x3x3 - All-Russian MO 1999 Regional (R4) 8.8

2 player domino game - All-Russian MO 1999 Regional (R4) 8.7

cannot divide 1-15 in 2 groups - All-Russian MO 1999 Regional (R4) 8.5

40 identical gas cylinders - All-Russian MO 1999 Regional (R4) 8.4

n=1+2+...+A - All-Russian MO 1999 Regional (R4) 8.2

father and two sons visit grandmother - All-Russian MO 1999 Regional (R4) 8.`

&lt;CA_1B=90^o, symmetric points wrt lines- All-Russian MO 1999 Regional (R4) 8.6

medians of inscribed triangle // sides - All-Russian MO 1999 Regional (R4) 8.3

|x+a|+|x+y+b|+|y+c|>|x|+|x+y|+|y| - All-Russian MO 1999 Regional (R4) 11.5

>=2 integer points for polynomial - All-Russian MO 1999 Regional (R4) 11.8

circumscribed polyhedron around a sphere has <= 6 large faces

sum 1/x^k >= sum x^k - All-Russian MO 1999 Regional (R4) 9.3

<CA_1B=90^o, symmetric points wrt lines- All-Russian MO 1999 Regional (R4) 8.6