MathDB

1

ASU 414 All Soviet Union MO 1985 x/(2+(x+(2+(x+(2+..))))=1

2

" encounters

1985

times):

\dfrac{x}{2+ \dfrac{x}{2+\dfrac{x}{2+... \dfrac{x}{2+\sqrt {1+x}}}}}=1

415

1

ASU 415 All Soviet Union MO 1985 regular pentagon remaining area

All the points situated more close than

1

cm to ALL the vertices of the regular pentagon with

1

cm side, are deleted from that pentagon. Find the area of the remained figure.

416

1

ASU 416 All Soviet Union MO 1985 unit rsquareas paper, rectangle area

Given big enough sheet of cross-lined paper with the side of the squares equal to

1

. We are allowed to cut it along the lines only. Prove that for every

m>12

we can cut out a rectangle of the greater than

m

area such, that it is impossible to cut out a rectangle of

m

area from it.

417

1

ASU 417 All Soviet Union MO 1985 circle's distance in different cube's planes

The

ABCDA_1B_1C_1D_1

cube has unit length edges. Find the distance between two circumferences, one of those is inscribed into the

ABCD

base, and another comes through points

A,C

and

B_1

.

413

1

ASU 413 All Soviet Union MO 1985 7 of 24 triangles in a hexagon

Given right hexagon. The lines parallel to all the sides are drawn from all the vertices and midpoints of the sides (consider only the interior, with respect to the hexagon, parts of those lines). Thus the hexagon is divided onto

24

triangles, and the figure has

19

nodes.

19

different numbers are written in those nodes. Prove that at least

7

of

24

triangles have the property: the numbers in its vertices increase (from the least to the greatest) counterclockwise.

412

1

ASU 412 All Soviet Union MO 1985 3 equal circumcircles

One of two circumferences of radius

R

comes through

A

and

B

vertices of the

ABCD

parallelogram. Another comes through

B

and

D

. Let

M

be another point of circumferences intersection. Prove that the circle circumscribed around

AMD

triangle has radius

R

.

411

1

ASU 411 All Soviet Union MO 1985 painting equal cubes of parallelepiped

The parallelepiped is constructed of the equal cubes. Three parallelepiped faces, having the common vertex are painted. Exactly half of all the cubes have at least one face painted. What is the total number of the cubes?

410

1

ASU 410 All Soviet Union MO 1985 sum of |a_i - b_i| = n^2

Numbers

1,2,3,...,2n

are divided onto two equal groups. Let

a_1,a_2,...,a_n

be the first group numbers in the increasing order, and

b_1,b_2,...,b_n

-- the second group numbers in the decreasing order. Prove that

|a_1 - b_1| + |a_2 - b_2| + ... + |a_n - b_n| = n^2

409

1

ASU 409 All Soviet Union MO 1985 (a,b,c,d) -> (a-b,b-c,c-d,d-a)

If there are four numbers

(a,b,c,d)

in four registers of the calculating machine, they turn into

(a-b,b-c,c-d,d-a)

numbers whenever you press the button. Prove that if not all the initial numbers are equal, machine will obtain at least one number more than

1985

after some number of the operations.

408

1

ASU 408 All Soviet Union MO 1985 |A_2A_3| + |MN| = circumradius

The

[A_0A_5]

diameter divides a circumference with the

O

centre onto two hemicircumferences. One of them is divided onto five equal arcs

A_0A_1, A_1A_2, A_2A_3, A_3A_4, A_4A_5

. The

(A_1A_4)

line crosses

(OA_2)

and

(OA_3)

lines in

M

and

N

points. Prove that

(|A_2A_3| + |MN|)

equals to the circumference radius.

407

1

ASU 407 All Soviet Union MO 1985 cube, cubic box, 6 colours, game

Given a cube, a cubic box, that exactly suits for the cube, and six colours. First man paints each side of the cube with its (side's) unique colour. Another man does the same with the box. Prove that the third man can put the cube in the box in such a way, that every cube side will touch the box side of different colour.

406

1

ASU 406 All Soviet Union MO 1985 no of painted parts by n straight lines

n

straight lines are drawn in a plane. They divide the plane onto several parts. Some of the parts are painted. Not a pair of painted parts has non-zero length common bound. Prove that the number of painted parts is not more than

\frac{n^2 + n}{3}

.

405

1

ASU 405 All Soviet Union MO 1985 a_{2n}=a_n, a_{4n+1}=1, a_{4n+3}=0

The sequence

a_1, a_2, ... , a_k, ...

is constructed according to the rules:

a_{2n} = a_n,a_{4n+1} = 1,a_{4n+3} = 0

Prove that it is non-periodical sequence.

404

1

ASU 404 All Soviet Union MO 1985 pentagon contruction, symmetry wrt point

The convex pentagon

ABCDE

was drawn in the plane.

A_1

was symmetric to

A

with respect to

B

.

B_1

was symmetric to

B

with respect to

C

.

C_1

was symmetric to

C

with respect to

D

.

D_1

was symmetric to

D

with respect to

E

.

E_1

was symmetric to

E

with respect to

A

. How is it possible to restore the initial pentagon with the compasses and ruler, knowing

A_1,B_1,C_1,D_1,E_1

points?

403

1

ASU 403 All Soviet Union MO 1985 sin(x)-sin(y)| + sin(x)sin(y)<= 0

Find all the pairs

(x,y)

such that

|\sin x-\sin y| + \sin x \sin y \le 0

.

402

1

ASU 402 All Soviet Union MO 1985 a_1/a_2+a_2/a_3+...+a_k/a_{k-1}<k-1985

Given unbounded strictly increasing sequence

a_1, a_2, ... , a_n, ...

of positive numbers. Prove that
a) there exists a number

k_0

such that for all

k>k_0

the following inequality is valid:

\frac{a_1}{a_2}+ \frac{a_2}{a_3} + ... + \frac{a_k}{a_{k-1} }< k - 1

b) there exists a number

k_0

such that for all

k>k_0

the following inequality is valid:

\frac{a_1}{a_2}+ \frac{a_2}{a_3} + ... + \frac{a_k}{a_{k-1} }< k - 1985

401

1

ASU 401 All Soviet Union MO 1985 like pascal triangle, min wanted

In the diagram below

a, b, c, d, e, f, g, h, i, j

are distinct positive integers and each (except

a, e, h

and

j

) is the sum of the two numbers to the left and above. For example,

b = a + e, f = e + h, i = h + j

. What is the smallest possible value of

d

?
j h i e f g a b c d

400

1

ASU 400 All Soviet Union MO 1985 max no integer x so |P(x)|<50, trinomial

The senior coefficient

a

in the square polynomial

P(x) = ax^2 + bx + c

is more than

100

. What is the maximal number of integer values of

x

, such that

|P(x)|<50

.

399

1

ASU 399 All Soviet Union MO 1985 rotations and symmetry

Given a straight line

\ell

and the point

O

out of the line. Prove that it is possible to move an arbitrary point

A

in the same plane to the

O

point, using only rotations around

O

and symmetry with respect to the

\ell

.

398

1

ASU 398 All Soviet Union MO 1985 painting all sides + diagonals of n-angle

You should paint all the sides and diagonals of the regular

n

-gon so, that every pair of segments, having the common point, would be painted with different colours. How many colours will you require?

397

1

ASU 397 All Soviet Union MO 1985 max no of men in 8x8 chessboard

What maximal number of the men in checkers game can be put on the chess-board

8\times 8

so, that every man can be taken by at least one other man ?

396

1

ASU 396 All Soviet Union MO 1985 sum of squares of digits 10^6

Is there any numbber

n

, such that the sum of its digits in the decimal notation is

1000

, and the sum of its square digits in the decimal notation is

1000000

?

395

1

ASU 395 All Soviet Union MO 1985 hexagon area is1/2 triangle area

Two perpendiculars are drawn from the midpoints of each side of the acute-angle triangle to two other sides. Those six segments make hexagon. Prove that the hexagon area is a half of the triangle area.

1985 All Soviet Union Mathematical Olympiad

Subcontests

ASU 414 All Soviet Union MO 1985 x/(2+(x+(2+(x+(2+..))))=1

ASU 415 All Soviet Union MO 1985 regular pentagon remaining area

ASU 416 All Soviet Union MO 1985 unit rsquareas paper, rectangle area

ASU 417 All Soviet Union MO 1985 circle's distance in different cube's planes

ASU 413 All Soviet Union MO 1985 7 of 24 triangles in a hexagon

ASU 412 All Soviet Union MO 1985 3 equal circumcircles

ASU 411 All Soviet Union MO 1985 painting equal cubes of parallelepiped

ASU 410 All Soviet Union MO 1985 sum of |a_i - b_i| = n^2

ASU 409 All Soviet Union MO 1985 (a,b,c,d) -> (a-b,b-c,c-d,d-a)

ASU 408 All Soviet Union MO 1985 |A_2A_3| + |MN| = circumradius

ASU 407 All Soviet Union MO 1985 cube, cubic box, 6 colours, game

ASU 406 All Soviet Union MO 1985 no of painted parts by n straight lines

ASU 405 All Soviet Union MO 1985 a_{2n}=a_n, a_{4n+1}=1, a_{4n+3}=0

ASU 404 All Soviet Union MO 1985 pentagon contruction, symmetry wrt point

ASU 403 All Soviet Union MO 1985 sin(x)-sin(y)| + sin(x)sin(y)<= 0

ASU 402 All Soviet Union MO 1985 a_1/a_2+a_2/a_3+...+a_k/a_{k-1}<k-1985

ASU 401 All Soviet Union MO 1985 like pascal triangle, min wanted

ASU 400 All Soviet Union MO 1985 max no integer x so |P(x)|<50, trinomial

ASU 399 All Soviet Union MO 1985 rotations and symmetry

ASU 398 All Soviet Union MO 1985 painting all sides + diagonals of n-angle

ASU 397 All Soviet Union MO 1985 max no of men in 8x8 chessboard

ASU 396 All Soviet Union MO 1985 sum of squares of digits 10^6

ASU 395 All Soviet Union MO 1985 hexagon area is1/2 triangle area

1985 All Soviet Union Mathematical Olympiad

Subcontests

ASU 414 All Soviet Union MO 1985 x/(2+(x+(2+(x+(2+..))))=1

ASU 415 All Soviet Union MO 1985 regular pentagon remaining area

ASU 416 All Soviet Union MO 1985 unit rsquareas paper, rectangle area

ASU 417 All Soviet Union MO 1985 circle&#039;s distance in different cube&#039;s planes

ASU 413 All Soviet Union MO 1985 7 of 24 triangles in a hexagon

ASU 412 All Soviet Union MO 1985 3 equal circumcircles

ASU 411 All Soviet Union MO 1985 painting equal cubes of parallelepiped

ASU 410 All Soviet Union MO 1985 sum of |a_i - b_i| = n^2

ASU 409 All Soviet Union MO 1985 (a,b,c,d) -&gt; (a-b,b-c,c-d,d-a)

ASU 408 All Soviet Union MO 1985 |A_2A_3| + |MN| = circumradius

ASU 407 All Soviet Union MO 1985 cube, cubic box, 6 colours, game

ASU 406 All Soviet Union MO 1985 no of painted parts by n straight lines

ASU 405 All Soviet Union MO 1985 a_{2n}=a_n, a_{4n+1}=1, a_{4n+3}=0

ASU 404 All Soviet Union MO 1985 pentagon contruction, symmetry wrt point

ASU 403 All Soviet Union MO 1985 sin(x)-sin(y)| + sin(x)sin(y)&lt;= 0

ASU 402 All Soviet Union MO 1985 a_1/a_2+a_2/a_3+...+a_k/a_{k-1}&lt;k-1985

ASU 401 All Soviet Union MO 1985 like pascal triangle, min wanted

ASU 400 All Soviet Union MO 1985 max no integer x so |P(x)|&lt;50, trinomial

ASU 399 All Soviet Union MO 1985 rotations and symmetry

ASU 398 All Soviet Union MO 1985 painting all sides + diagonals of n-angle

ASU 397 All Soviet Union MO 1985 max no of men in 8x8 chessboard

ASU 396 All Soviet Union MO 1985 sum of squares of digits 10^6

ASU 395 All Soviet Union MO 1985 hexagon area is1/2 triangle area

ASU 417 All Soviet Union MO 1985 circle's distance in different cube's planes

ASU 409 All Soviet Union MO 1985 (a,b,c,d) -> (a-b,b-c,c-d,d-a)

ASU 403 All Soviet Union MO 1985 sin(x)-sin(y)| + sin(x)sin(y)<= 0

ASU 402 All Soviet Union MO 1985 a_1/a_2+a_2/a_3+...+a_k/a_{k-1}<k-1985

ASU 400 All Soviet Union MO 1985 max no integer x so |P(x)|<50, trinomial