MathDB

1

ASU 349 All Soviet Union MO 1983 sets in 4x4 unit grid

4\times 4

square grid net, has

1\times 1

size. Is it possible to represent this net as a union of the following sets:
a) Eight broken lines of length five each?
b) Five broken lines of length eight each?

370

1

ASU 370 All Soviet Union MO 1983 no of diff.segments in decimal repr.

The infinite decimal notation of the real number

x

contains all the digits. Let

u_n

be the number of different

n

-digit segments encountered in

x

notation. Prove that if for some

n

,

u_n \le (n+8)

, than

x

is a rational number.

369

1

ASU 369 All Soviet Union MO 1983 k non-intersecting segments in set M

The

M

set consists of

k

non-intersecting segments on the line. It is possible to put an arbitrary segment shorter than

1

cm on the line in such a way, that his ends will belong to

M

. Prove that the total sum of the segment lengths is not less than

1/k

cm.

368

1

ASU 368 All Soviet Union MO 1983 maximal sidelength inequality

The points

D,E,F

belong to the sides

(AB), (BC)

and

(CA)

of the triangle

ABC

respectively (but they are not vertices). Let us denote with

d_0, d_1, d_2

, and

d_3

the maximal side length of the triangles

DEF

,

DEA

,

DBF

,

CEF

, respectively. Prove that

d_0 \ge \frac{\sqrt3}{2} min\{d_1, d_2, d_3\}

When the equality takes place?

367

1

ASU 367 All Soviet Union MO 1983 2m+1 integers with abs.value <=2m-1

Given

(2m+1)

different integers, each absolute value is not greater than

(2m-1)

. Prove that it is possible to choose three numbers among them, with their sum equal to zero.

366

1

ASU 366 All Soviet Union MO 1983 Caratheodory's theorem with vectors and areas

Given a point

O

inside triangle

ABC

. Prove that

S_A * \overrightarrow{OA} + S_B * \overrightarrow{OB} + S_C * \overrightarrow{OC} = \overrightarrow{0}

where

S_A, S_B, S_C

denote areas of triangles

BOC, COA, AOB

respectively.

365

1

ASU 365 All Soviet Union MO 1983 4 rectangles inside rectangle, area inequality

One side of the rectangle is

1

cm. It is known that the rectangle can be divided by two orthogonal lines onto four rectangles, and each of the smaller rectangles has the area not less than

1

square centimetre, and one of them is not less than

2

square centimetres. What is the least possible length of another side of big rectangle?

364

1

ASU 364 All Soviet Union MO 1983 kindergarten children's no divisible by 8

The kindergarten group is standing in the column of pairs. The number of boys equals the number of girls in each of the two columns. The number of mixed (boy and girl) pairs equals to the number of the rest pairs. Prove that the total number of children in the group is divisible by eight.

363

1

ASU 363 All Soviet Union MO 1983 equal areas inside a triangle

The points

A_1,B_1,C_1

belong to

[BC],[CA],[AB]

sides of the

ABC

triangle respectively. The

[AA_1], [BB_1], [CC_1]

segments split the

ABC

onto

4

smaller triangles and

3

quadrangles. It is known, that the smaller triangles have the same area. Prove that the quadrangles have equal areas. What is the quadrangle area, it the small triangle has the unit area?

362

1

ASU 362 All Soviet Union MO 1983 squares of infinite cross-lined paper with sum

Can You fill the squares of the infinite cross-lined paper with integers so, that the sum of the numbers in every

4\times 6

fields rectangle would be
a)

10

?
b)

1

?

361

1

ASU 361 All Soviet Union MO 1983 tribe language alphabet contains 2 letters

The Abba tribe language alphabet contains two letters only. Not a word of this language is a beginning of another word. Can this tribe vocabulary contain

3

four-letter,

10

five-letter,

30

six-letter and

5

seven-letter words?

360

1

ASU 360 All Soviet Union MO 1983 n^m / m^n and k^n / n^k =>k^m / m^k

Given natural

n,m,k

. It is known that

m^n

is divisible by

n^m

, and

n^k

is divisible by

k^n

. Prove that

m^k

is divisible by

k^m

.

359

1

ASU 359 All Soviet Union MO 1983 training in square equation solution

The pupil is training in the square equation solution. Having the recurrent equation solved, he stops, if it doesn't have two roots, or solves the next equation, with the free coefficient equal to the greatest root, the coefficient at

x

equal to the least root, and the coefficient at

x^2

equal to

1

. Prove that the process cannot be infinite. What maximal number of the equations he will have to solve?

358

1

ASU 358 All Soviet Union MO 1983 // planes, projections of tetrahedron

The points

A_1,B_1,C_1,D_1

and

A_2,B_2,C_2,D_2

are orthogonal projections of the

ABCD

tetrahedron vertices on two planes. Prove that it is possible to move one of the planes to provide the parallelness of lines

(A_1A_2), (B_1B_2), (C_1C_2)

and

(D_1D_2)

.

357

1

ASU 357 All Soviet Union MO 1983 sin^2a+sin^2a = sin(a+b)

Two acute angles

a

and

b

satisfy condition

\sin^2a+\sin^2b = \sin(a+b)

Prove that

a + b = \pi /2

.

356

1

ASU 356 All Soviet Union MO 1983 a_n last digits of [\sqrt{10}^n], periodical

The sequences

a_n

and

b_n

members are the last digits of

[\sqrt{10}^n]

and

[\sqrt{2}^n]

respectively (here

[ ...]

denotes the whole part of a number). Are those sequences periodical?

355

1

ASU 355 All Soviet Union MO 1983 (DEF)<= (ADE)+(BDF) areas

The point

D

is the midpoint of the side

[AB]

of the triangle

ABC

. The points

E

and

F

belong to

[AC]

and

[BC]

respectively. Prove that the area of triangle

DEF

area does not exceed the sum of the areas of triangles

ADE

and

BDF

.

354

1

ASU 354 All Soviet Union MO 1983 inequality in rounding in tens

Natural number

k

has

n

digits in its decimal notation. It was rounded up to tens, then the obtained number was rounded up to hundreds, and so on

(n-1)

times. Prove that the obtained number

m

satisfies inequality

m < \frac{18k}{13}

. (Examples of rounding:

191\to190\to 200, 135\to140\to 100

.)

352

1

ASU 353 All Soviet Union MO 1983 y^2=x^3-3x^2 + 2x and x^2=y^3-3y^2+2y

Find all the solutions of the system

\begin{cases} y^2 = x^3 - 3x^2 + 2x \\ x^2 = y^3 - 3y^2 + 2y \end{cases}

351

1

ASU 351 All Soviet Union MO 1983 expand disks by 2/\sqrt3 times

Three disks touch pairwise from outside in the points

X,Y,Z

. Then the radiuses of the disks were expanded by

2/\sqrt3

times, and the centres were reserved. Prove that the triangle

XYZ

is completely covered by the expanded disks.

350

1

ASU 350 All Soviet Union MO1983 3 numbers, blackboard {17, 1967, 1983}

Three numbers were written with a chalk on the blackboard. The following operation was repeated several times: One of the numbers was cleared and the sum of two other numbers, decreased by

1

, was written instead of it. The final set of numbers is

\{17, 1967, 1983\}

.Is it possible to admit that the initial numbers were
a)

\{2, 2, 2\}

?
b)

\{3, 3, 3\}

?

1983 All Soviet Union Mathematical Olympiad

Subcontests

ASU 349 All Soviet Union MO 1983 sets in 4x4 unit grid

ASU 370 All Soviet Union MO 1983 no of diff.segments in decimal repr.

ASU 369 All Soviet Union MO 1983 k non-intersecting segments in set M

ASU 368 All Soviet Union MO 1983 maximal sidelength inequality

ASU 367 All Soviet Union MO 1983 2m+1 integers with abs.value <=2m-1

ASU 366 All Soviet Union MO 1983 Caratheodory's theorem with vectors and areas

ASU 365 All Soviet Union MO 1983 4 rectangles inside rectangle, area inequality

ASU 364 All Soviet Union MO 1983 kindergarten children's no divisible by 8

ASU 363 All Soviet Union MO 1983 equal areas inside a triangle

ASU 362 All Soviet Union MO 1983 squares of infinite cross-lined paper with sum

ASU 361 All Soviet Union MO 1983 tribe language alphabet contains 2 letters

ASU 360 All Soviet Union MO 1983 n^m / m^n and k^n / n^k =>k^m / m^k

ASU 359 All Soviet Union MO 1983 training in square equation solution

ASU 358 All Soviet Union MO 1983 // planes, projections of tetrahedron

ASU 357 All Soviet Union MO 1983 sin^2a+sin^2a = sin(a+b)

ASU 356 All Soviet Union MO 1983 a_n last digits of [\sqrt{10}^n], periodical

ASU 355 All Soviet Union MO 1983 (DEF)<= (ADE)+(BDF) areas

ASU 354 All Soviet Union MO 1983 inequality in rounding in tens

ASU 353 All Soviet Union MO 1983 y^2=x^3-3x^2 + 2x and x^2=y^3-3y^2+2y

ASU 351 All Soviet Union MO 1983 expand disks by 2/\sqrt3 times

ASU 350 All Soviet Union MO1983 3 numbers, blackboard {17, 1967, 1983}

1983 All Soviet Union Mathematical Olympiad

Subcontests

ASU 349 All Soviet Union MO 1983 sets in 4x4 unit grid

ASU 370 All Soviet Union MO 1983 no of diff.segments in decimal repr.

ASU 369 All Soviet Union MO 1983 k non-intersecting segments in set M

ASU 368 All Soviet Union MO 1983 maximal sidelength inequality

ASU 367 All Soviet Union MO 1983 2m+1 integers with abs.value &lt;=2m-1

ASU 366 All Soviet Union MO 1983 Caratheodory&#039;s theorem with vectors and areas

ASU 365 All Soviet Union MO 1983 4 rectangles inside rectangle, area inequality

ASU 364 All Soviet Union MO 1983 kindergarten children's no divisible by 8

ASU 363 All Soviet Union MO 1983 equal areas inside a triangle

ASU 362 All Soviet Union MO 1983 squares of infinite cross-lined paper with sum

ASU 361 All Soviet Union MO 1983 tribe language alphabet contains 2 letters

ASU 360 All Soviet Union MO 1983 n^m / m^n and k^n / n^k =&gt;k^m / m^k

ASU 359 All Soviet Union MO 1983 training in square equation solution

ASU 358 All Soviet Union MO 1983 // planes, projections of tetrahedron

ASU 357 All Soviet Union MO 1983 sin^2a+sin^2a = sin(a+b)

ASU 356 All Soviet Union MO 1983 a_n last digits of [\sqrt{10}^n], periodical

ASU 355 All Soviet Union MO 1983 (DEF)&lt;= (ADE)+(BDF) areas

ASU 354 All Soviet Union MO 1983 inequality in rounding in tens

ASU 353 All Soviet Union MO 1983 y^2=x^3-3x^2 + 2x and x^2=y^3-3y^2+2y

ASU 351 All Soviet Union MO 1983 expand disks by 2/\sqrt3 times

ASU 350 All Soviet Union MO1983 3 numbers, blackboard {17, 1967, 1983}

ASU 367 All Soviet Union MO 1983 2m+1 integers with abs.value <=2m-1

ASU 366 All Soviet Union MO 1983 Caratheodory's theorem with vectors and areas

ASU 360 All Soviet Union MO 1983 n^m / m^n and k^n / n^k =>k^m / m^k

ASU 355 All Soviet Union MO 1983 (DEF)<= (ADE)+(BDF) areas