MathDB

1

is pentagon convex? - All-Russian MO 1998 Regional (R4) 10.6

A_1A_2A_3A_4A_5

contains bisectors

\ell_1

,

\ell_2

,

...

,

\ell_5

of angles

\angle A_1

,

\angle A_2

,

...

,

\angle A_5

respectively. Bisectors

\ell_1

and

\ell_2

intersect at point

B_1

,

\ell_2

and

\ell_3

- at point

B_2

, etc.,

\ell_5

and

\ell_1

intersect at point

B_5

. Can the pentagon

B_1B_2B_3B_4B_5

be convex?

11.7

1

volo f set of midpoints of segments of 2 central symmetric reg. tetrahedra

Given two regular tetrahedrons with edges of length

\sqrt2

, transforming into one another with central symmetry. Let

\Phi

be the set the midpoints of segments whose ends belong to different tetrahedrons. Find the volume of the figure

\Phi

.

11.6

1

A-B = 4(a -b) bw cells - All-Russian MO 1998 Regional (R4) 11.6

A polygon with sides running along the sides of the squares was cut out of an endless chessboard. A segment of the perimeter of a polygon is called black if the polygon adjacent to it from the inside is which cell is black, respectively white if the cell is white. Let

A

be the number of black segments on the perimeter, and

B

be the number of white ones, Let the polygon consist of

a

black and

b

white cells. Prove that

A-B = 4(a -b)

.

11.5

1

7^{1998}. -> 1998^7 - All-Russian MO 1998 Regional (R4) 11.5

A whole number is written on the board. Its last digit is remembered is then erased and multiplied by

5

added to the number that remained on the board after erasing. The number was originally written

7^{1998}

. After applying several such operations, can one get the number

1998^7

?

11.4

1

n-1 cells of 0 and 1 in n x n table - All-Russian MO 1998 Regional (R4) 11.4

There is an

n \times n

table with

n -1

cells containing ones and the remaining cells containing zeros. You can do this with the table the following operation: select the tap hole, subtract from the number in this cell, one, and to all other numbers on the same line or in the same column as the selected cell, add one. Is it possible from of this table, using the specified operations, obtain a table in which all numbers are equal?

11.2

1

DE _|_ OC, interssecting circles

Circle

S

with center

O

and circle

S'

intersect at points

A

and

B

. Point

C

is taken on the arc of a circle

S

lying inside

S'

. Denote the intersection points of

AC

and

BC

with

S'

, other than

A

and

B

, as

E

and

D

, respectively. Prove that lines

DE

and

OC

are perpendicular.

10.8

1

money to guess no from 1 to 144 - All-Russian MO 1998 Regional (R4) 10.8

A number from

1

to

144

is guessed. You are allowed to select a subset of the set of numbers from

1

to

144

and ask whether the guessed number belongs to it. For the answer “yes” you have to pay

2

rubles, for the answer “no” -

1

ruble. What is the smallest amount of money needed to surely guess that?

10.5

1

{ (x + 1)^3 } = x^3 - All-Russian MO 1998 Regional (R4) 10.5

Solve the equation

\{(x + 1)^3\} = x^3

, where

\{z\}

is the fractional part of the number z, i.e.

\{z\} = z - [z]

.

10.4

1

numbers 1-2^{1998} in 1999 cells {- All-Russian MO 1998 Regional (R4) 10.4

In the first

1999

cells of the computer are written numbers in the specified order::

1

,

2

,

4

,

...

,

2^{1998}

. Two programmers take turns reducing in one move per unit number in five different cells. If a negative number appears in one of the cells, then the computer breaks down and the broken repairs are paid for. Which programmer can protect himself from financial losses, regardless of his partner’s moves, and how should he do this act?

10.3

1

n-1 points may be be split into two parts of smaller diameter.

Prove that from any finite set of points on the plane, you can remove a point from the bottom in such a way that the remaining set can be split into two parts of smaller diameter. (Diameter is the maximum distance between points of the set.)
[hide=original wording]Докажите, что из любого конечного множества точек на плоскости можно так удалитьо дну точку, что оставшееся множество можно разбить на две части меньшего диаметра. (Диаметр—это максимальное расстояние между точками множества.)

10.2

1

parallelogram wanted, (BOC) related

In an acute triangle

ABC

, a circle

S

is drawn through the center

O

of the circumcircle and the vertices

B

and

C

. Let

OK

be the diameter of the circle

S

,

D

and

E

, be it's intersection points with the straight lines

AB

and

AC

respectively. Prove that

ADKE

is a parallelogram.

10.1

1

f(x) = x^2 + ax + b cos x - All-Russian MO 1998 Regional (R4) 10.1

Let

f(x) = x^2 + ax + b cos x

. Find all values of parameter

a

and

b

, for which the equations

f(x) = 0

and

f(f(x)) = 0

have the same non-empty sets of real roots.

9.7

1

regular 1998-gon billiard - All-Russian MO 1998 Regional (R4) 9.7

Given a billiard in the form of a regular

1998

-gon

A_1A_2...A_{1998}

. A ball was released from the midpoint of side

A_1A_2

, which, reflected therefore from sides

A_2A_3

,

A_3A_4

, . . . ,

A_{1998}A_1

(according to the law, the angle of incidence is equal to the angle of reflection), returned to the starting point. Prove that the trajectory of the ball is a regular

1998

-gon.

9.6

1

2 chips in 1x101 - All-Russian MO 1998 Regional (R4) 9.6

At the ends of a checkered strip measuring

1 \times 101

squares there are two chips: on the left is the chip of the first player, on the right is the second. Per turn dares to move his piece in the direction of the opposite edge of the strip by 1, 2, 3 or 4 cells. In this case, you are allowed to jump over opponent's chip, but it is forbidden to place your chip on the same square with her. The first one to reach the opposite edge of the strip wins. Who wins if the game is played correctly: the one who goes first, or him rival?

9.5

1

2 trinomials - All-Russian MO 1998 Regional (R4) 9.5

The roots of the two monic square trinomials are negative integers, and one of these roots is common. Can the values of these trinomials at some positive integer point equal 19 and 98?

9.4

1

101 cels in 102x102 - All-Russian MO 1998 Regional (R4) 9.4

There is a square of checkered paper measuring

102 \times 102

squares and a connected figure of unknown shape, consisting of 101 cells. What is the largest number of such figures that can be cut from this square with a guarantee? A figure made up of cells is called connected if any two its cells can be connected by a chain of its cells in which any two adjacent cells have a common side.

9.2

1

<APB =<CQD, intersecting circles - All-Russian MO 1998 Regional (R4) 9.2

Two circles intersect at points

P

and

Q

. The straight line intersects these circles at points

A

,

B

,

C

,

D

, as shown in fig. . Prove that

\angle APB = \angle CQD

. https://cdn.artofproblemsolving.com/attachments/1/a/a581e11be68bbb628db5b5b8e75c7ff6e196c5.png

9.1

1

a,b,c,2r in arithm. progression - All-Russian MO 1998 Regional (R4) 9.1

The lengths of the sides of a certain triangle and the diameter of the inscribed part circles are four consecutive terms of arithmetic progression. Find all such triangles.

8.8

1

elections and polls - All-Russian MO 1998 Regional (R4) 8.8

In elections to the City Duma, each voter, if he goes to the polls, casts a vote for himself (if he is a candidate) and for those candidates who are his friends. The forecast of the sociological service of the mayor's office is considered good if it correctly predicts the number of votes cast for at least one of the candidates, and bad otherwise. Prove that for any forecast, voters can turn out to vote in such a way that this forecast turns out to be bad.

8.7

1

sum of 3 angles = 180^o, 3 circles - All-Russian MO 1998 Regional (R4) 8.7

Let

O

be the center of a circle circumscribed about an acute angle triangle

ABC

,

S_A

,

S_B

,

S_C

- circles with center O, tangent to sides

BC

,

CA

,

AB

respectively. Prove that the sum of three angles : between the tangents to

S_A

drawn from point

A

, to

S_B

from point

B

and to

S_C

- from point

C

, is equal to

180^o

.

8.6

1

128 sheep - All-Russian MO 1998 Regional (R4) 8.6

Several farmers have 128 sheep. If one of them has at least half of all sheep, the rest conspire and dispossess him: everyone takes as many sheep as he already has : If two people have 64 sheep, then one of them is dispossessed. There were 7 dispossessions. Prove that all the sheep were gathered from one peasant.

8.5

1

numbers 1-9 in a figure - All-Russian MO 1998 Regional (R4) 8.5

Place numbers from

1

to

9

in the circles of the figure (see Fig. ) so that the sum of four numbers, finding located in the circles at the tops of all squares (there are six of them), was constant , https://cdn.artofproblemsolving.com/attachments/8/8/5fe1e8c5949903dd9500b992c8139277cebe7f.png

8.4

1

n points lie on 2 circles - All-Russian MO 1998 Regional (R4) 8.4

A set of

n\ge 9

points is given on the plane. For any 9 it points can be selected from all circles so that all these points end up on selected circles. Prove that all n points lie on two circles

8.3

1

52 cards in the deck - All-Russian MO 1998 Regional (R4) 8.3

There are 52 cards in the deck, 13 of each suit. Vanya draws from the deck one card at a time. Removed cards are not returned to the deck. Every time Before taking out the card, Vanya makes a wish for some suit.Prove that if Vanya makes a wish every time, , the cards of which are in the deck has no less cards left than cards of any other suit, then the hidden suit will fall with the suit of the card drawn at least 13 times.

8.1

1

n-digit numbers M , N - All-Russian MO 1998 Regional (R4) 8.1

Are there

n

-digit numbers M and N such that all digits

M

are even, all

N

digits are odd, every digit from

0

to

9

occurs in decimal notation M or N at least once, and

M

is divisible by

N

?

10.7

1

Nice and easy combinatorics

A cube of side length

n

is divided into unit cubes by partitions (each partition separates a pair of adjacent unit cubes). What is the smallest number of partitions that can be removed so that from each cube, one can reach the surface of the cube without passing through a partition ?

11.8

1

Tricky combo, and the need to eliminate m.

A sequence

a_1,a_2,\cdots

of positive integers contains each positive integer exactly once. Moreover for every pair of distinct positive integer

m

and

n

,

\frac{1}{1998} < \frac{|a_n- a_m|}{|n-m|} < 1998

, show that

|a_n - n | <2000000

for all

n

.

11.1

1

Basic Summations

Two identical decks have 36 cards each. One deck is shuffled and put on top of the second. For each card of the top deck, we count the number of cards between it and the corresponding card of the bottom deck. What is the sum of these numbers?
Sorry if this has been posted before but I would like to know if I solved it correctly. Thanks!

9.8

1

Russian Combinatorics - Compass in grid

The endpoints of a compass are at two lattice points of an inﬁnite unit square grid. It is allowed to rotate the compass around one of its endpoints, not varying its radius, and thus move the other endpoint to another lattice point. Can the endpoints of the compass change places after several such steps?

8.2

1

Interesting [can the lines AM and AN trisect < BAD ?]

Given a parallelogram ABCD, let M and N be the midpoints of the sides BC and CD. Can the lines AM, AN divide the angle BAD into three equal angles?

1998 All-Russian Olympiad Regional Round

Subcontests

is pentagon convex? - All-Russian MO 1998 Regional (R4) 10.6

volo f set of midpoints of segments of 2 central symmetric reg. tetrahedra

A-B = 4(a -b) bw cells - All-Russian MO 1998 Regional (R4) 11.6

7^{1998}. -> 1998^7 - All-Russian MO 1998 Regional (R4) 11.5

n-1 cells of 0 and 1 in n x n table - All-Russian MO 1998 Regional (R4) 11.4

DE _|_ OC, interssecting circles

money to guess no from 1 to 144 - All-Russian MO 1998 Regional (R4) 10.8

{ (x + 1)^3 } = x^3 - All-Russian MO 1998 Regional (R4) 10.5

numbers 1-2^{1998} in 1999 cells {- All-Russian MO 1998 Regional (R4) 10.4

n-1 points may be be split into two parts of smaller diameter.

parallelogram wanted, (BOC) related

f(x) = x^2 + ax + b cos x - All-Russian MO 1998 Regional (R4) 10.1

regular 1998-gon billiard - All-Russian MO 1998 Regional (R4) 9.7

2 chips in 1x101 - All-Russian MO 1998 Regional (R4) 9.6

2 trinomials - All-Russian MO 1998 Regional (R4) 9.5

101 cels in 102x102 - All-Russian MO 1998 Regional (R4) 9.4

<APB =<CQD, intersecting circles - All-Russian MO 1998 Regional (R4) 9.2

a,b,c,2r in arithm. progression - All-Russian MO 1998 Regional (R4) 9.1

elections and polls - All-Russian MO 1998 Regional (R4) 8.8

sum of 3 angles = 180^o, 3 circles - All-Russian MO 1998 Regional (R4) 8.7

128 sheep - All-Russian MO 1998 Regional (R4) 8.6

numbers 1-9 in a figure - All-Russian MO 1998 Regional (R4) 8.5

n points lie on 2 circles - All-Russian MO 1998 Regional (R4) 8.4

52 cards in the deck - All-Russian MO 1998 Regional (R4) 8.3

n-digit numbers M , N - All-Russian MO 1998 Regional (R4) 8.1

Nice and easy combinatorics

Tricky combo, and the need to eliminate m.

Basic Summations

Russian Combinatorics - Compass in grid

Interesting [can the lines AM and AN trisect < BAD ?]

1998 All-Russian Olympiad Regional Round

Subcontests

is pentagon convex? - All-Russian MO 1998 Regional (R4) 10.6

volo f set of midpoints of segments of 2 central symmetric reg. tetrahedra

A-B = 4(a -b) bw cells - All-Russian MO 1998 Regional (R4) 11.6

7^{1998}. -&gt; 1998^7 - All-Russian MO 1998 Regional (R4) 11.5

n-1 cells of 0 and 1 in n x n table - All-Russian MO 1998 Regional (R4) 11.4

DE _|_ OC, interssecting circles

money to guess no from 1 to 144 - All-Russian MO 1998 Regional (R4) 10.8

{ (x + 1)^3 } = x^3 - All-Russian MO 1998 Regional (R4) 10.5

numbers 1-2^{1998} in 1999 cells {- All-Russian MO 1998 Regional (R4) 10.4

n-1 points may be be split into two parts of smaller diameter.

parallelogram wanted, (BOC) related

f(x) = x^2 + ax + b cos x - All-Russian MO 1998 Regional (R4) 10.1

regular 1998-gon billiard - All-Russian MO 1998 Regional (R4) 9.7

2 chips in 1x101 - All-Russian MO 1998 Regional (R4) 9.6

2 trinomials - All-Russian MO 1998 Regional (R4) 9.5

101 cels in 102x102 - All-Russian MO 1998 Regional (R4) 9.4

&lt;APB =&lt;CQD, intersecting circles - All-Russian MO 1998 Regional (R4) 9.2

a,b,c,2r in arithm. progression - All-Russian MO 1998 Regional (R4) 9.1

elections and polls - All-Russian MO 1998 Regional (R4) 8.8

sum of 3 angles = 180^o, 3 circles - All-Russian MO 1998 Regional (R4) 8.7

128 sheep - All-Russian MO 1998 Regional (R4) 8.6

numbers 1-9 in a figure - All-Russian MO 1998 Regional (R4) 8.5

n points lie on 2 circles - All-Russian MO 1998 Regional (R4) 8.4

52 cards in the deck - All-Russian MO 1998 Regional (R4) 8.3

n-digit numbers M , N - All-Russian MO 1998 Regional (R4) 8.1

Nice and easy combinatorics

Tricky combo, and the need to eliminate m.

Basic Summations

Russian Combinatorics - Compass in grid

Interesting [can the lines AM and AN trisect &lt; BAD ?]

7^{1998}. -> 1998^7 - All-Russian MO 1998 Regional (R4) 11.5

<APB =<CQD, intersecting circles - All-Russian MO 1998 Regional (R4) 9.2

Interesting [can the lines AM and AN trisect < BAD ?]