MathDB

1

ASU 303 All Soviet Union MO 1980 rearranging 5 digits into a sequence, [0,1]

x

from

[0,1]

is written as an infinite decimal fraction. Having rearranged its first five digits after the point we can obtain another fraction that corresponds to the number

x_1

. Having rearranged five digits of

x_k

from

(k+1)

-th till

(k+5)

-th after the point we obtain the number

x_{k+1}

.
a) Prove that the sequence

x_i

has limit.
b) Can this limit be irrational if we have started with the rational number?
c) Invent such a number, that always produces irrational numbers, no matter what digits were transposed.

302

1

ASU 302 All Soviet Union MO 1980 tetrahedron , cos (AC,BD) <= |CD|/|AB|

The edge

[AC]

of the tetrahedron

ABCD

is orthogonal to

[BC]

, and

[AD]

is orthogonal to

[BD]

. Prove that the cosine of the angle between lines

(AC)

and

(BD)

is less than

|CD|/|AB|

.

301

1

ASU 301 All Soviet Union MO 1980 [x^3/2]r +[y^3/2]=B has 1980 int. solutions

Prove that there is an infinite number of such numbers

B

that the equation

\lfloor x^3/2\rfloor + \lfloor y^3/2 \rfloor = B

has at least

1980

integer solutions

(x,y)

.
(

\lfloor z\rfloor

denotes the greatest integer not exceeding

z

.)

300

1

ASU 300 All Soviet Union MO 1980 integer set with min 1, max 100

The

A

set consists of integers only. Its minimal element is

1

and its maximal element is

100

. Every element of

A

except

1

equals to the sum of two (may be equal) numbers being contained in

A

. What is the least possible number of elements in

A

?

299

1

ASU 299 All Soviet Union MO 1980 x<1/3(p/4-sqrt(d^2-s/2)) in parallelepiped

Let the edges of rectangular parallelepiped be

x,y

and

z

(

x<y<z

). Let

p=4(x+y+z), s=2(xy+yz+zx) \,\,\, and \,\,\, d=\sqrt{x^2+y^2+z^2}

be its perimeter, surface area and diagonal length, respectively. Prove that

x < \frac{1}{3}\left( \frac{p}{4}- \sqrt{d^2 - \frac{s}{2}}\right )\,\,\, and \,\,\, z > \frac{1}{3}\left( \frac{p}{4}- \sqrt{d^2 - \frac{s}{2}}\right )

298

1

ASU 298 All Soviet Union MO 1980 angle wanted, equilateral starting

Given equilateral triangle

ABC

. Some line, parallel to

[AC]

crosses

[AB]

and

[BC]

in

M

and

P

points respectively. Let

D

be the centre of

PMB

triangle,

E

be the midpoint of the

[AP]

segment. Find the angles of triangle

DEC

.

297

1

ASU 297 All Soviet Union MO 1980 n_{k+1} = n_k + P(n_k), product of digits

Let us denote with

P(n)

the product of all the digits of

n

. Consider the sequence

n_{k+1} = n_k + P(n_k)

Can it be unbounded for some

n_1

?

296

1

ASU 296 All Soviet Union MO 1980 epidemic influenza and elfs

An epidemic influenza broke out in the elves city. First day some of them were infected by the external source of infection and nobody later was infected by the external source. The elf is infected when visiting his ill friend. In spite of the situation every healthy elf visits all his ill friends every day. The elf is ill one day exactly, and has the immunity at least on the next day. There is no graftings in the city. Prove that a) If there were some elves immunised by the external source on the first day, the epidemic influenza can continue arbitrary long time. b) If nobody had the immunity on the first day, the epidemic influenza will stop some day.

295

1

ASU 295 All Soviet Union MO 1980 red squares in infinite sheet of paper

Some squares of the infinite sheet of cross-lined paper are red. Each

2\times 3

rectangle (of

6

squares) contains exactly two red squares. How many red squares can be in the

9\times 11

rectangle of

99

squares?

294

1

ASU 294 All Soviet Union MO 1980 n+S(n)=1980, sum of digits

Let us denote with

S(n)

the sum of all the digits of

n

.
a) Is there such an

n

that

n+S(n)=1980

?
b) Prove that at least one of two arbitrary successive natural numbers is representable as

n + S(n)

for some third number

n

.

293

1

ASU 293 All Soviet Union MO 1980 1980 vectors in plane

Given

1980

vectors in the plane, and there are some non-collinear among them. The sum of every

1979

vectors is collinear to the vector not included in that sum. Prove that the sum of all vectors equals to the zero vector.

292

1

ASU 292 All Soviet Union MO 1980 trigonometric system of equation

Find real solutions of the system :

\begin{cases} \sin x + 2 \sin (x+y+z) = 0 \\ \sin y + 3 \sin (x+y+z) = 0\\ \sin z + 4 \sin (x+y+z) = 0\end{cases}

291

1

ASU 291 All Soviet Union MO 1980 dividible by 37

The six-digit decimal number contains six different non-zero digits and is divisible by

37

. Prove that having transposed its digits you can obtain at least

23

more numbers divisible by

37

290

1

ASU 290 All Soviet Union MO 1980 settlements on bank of Big Round Lake

There are several settlements on the bank of the Big Round Lake. Some of them are connected with the regular direct ship lines. Two settlements are connected if and only if two next (counterclockwise) to each ones are not connected. Prove that you can move from the arbitrary settlement to another arbitrary settlement, having used not more than three ships.

289

1

ASU 289 All Soviet Union MO 1980 chord that maximise the area of ABCD

Given a point

E

on the diameter

AC

of the certain circle. Draw a chord

BD

to maximise the area of the quadrangle

ABCD

.

288

1

ASU 288 All Soviet Union MO 1980 diophantine x^2 + y^3 = z^4

Are there three integers

x,y,z

, such that

x^2 + y^3 = z^4

?

287

1

ASU 287 All Soviet Union MO 1980 |AM| + |AP| = a, area of ABCD < a^2 /2

The points

M

and

P

are the midpoints of

[BC]

and

[CD]

sides of a convex quadrangle

ABCD

. It is known that

|AM| + |AP| = a

. Prove that

ABCD

has area less than

\frac{a^2}{2}

.

286

1

ASU 286 All Soviet Union MO 1980 35 containers, 3 metric tons each, 18 total

The load for the space station "Salute" is packed in containers. There are more than

35

containers, and the total weight is

18

metric tons. There are

7

one-way transport spaceships "Progress", each able to bring

3

metric tons to the station. It is known that they are able to take an arbitrary subset of

35

containers. Prove that they are able to take all the load.

285

1

ASU 285 All Soviet Union MO 1980 strips inside a square

The vertical side of a square is divided onto

n

segments. The sum of the segments with even numbers lengths equals to the sum of the segments with odd numbers lengths.

n-1

lines parallel to the horizontal sides are drawn from the segments ends, and, thus,

n

strips are obtained. The diagonal is drawn from the lower left corner to the upper right one. This diagonal divides every strip onto left and right parts. Prove that the sum of the left parts of odd strips areas equals to the sum of the right parts of even strips areas.

284

1

ASU 284 All Soviet Union MO 1980 all numbers from 19 to 80, divisible by 1980

All the two-digit numbers from

19

to

80

are written in a line without spaces. Is the obtained number

192021....7980

divisible by

1980

?

1980 All Soviet Union Mathematical Olympiad

Subcontests

ASU 303 All Soviet Union MO 1980 rearranging 5 digits into a sequence, [0,1]

ASU 302 All Soviet Union MO 1980 tetrahedron , cos (AC,BD) <= |CD|/|AB|

ASU 301 All Soviet Union MO 1980 [x^3/2]r +[y^3/2]=B has 1980 int. solutions

ASU 300 All Soviet Union MO 1980 integer set with min 1, max 100

ASU 299 All Soviet Union MO 1980 x<1/3(p/4-sqrt(d^2-s/2)) in parallelepiped

ASU 298 All Soviet Union MO 1980 angle wanted, equilateral starting

ASU 297 All Soviet Union MO 1980 n_{k+1} = n_k + P(n_k), product of digits

ASU 296 All Soviet Union MO 1980 epidemic influenza and elfs

ASU 295 All Soviet Union MO 1980 red squares in infinite sheet of paper

ASU 294 All Soviet Union MO 1980 n+S(n)=1980, sum of digits

ASU 293 All Soviet Union MO 1980 1980 vectors in plane

ASU 292 All Soviet Union MO 1980 trigonometric system of equation

ASU 291 All Soviet Union MO 1980 dividible by 37

ASU 290 All Soviet Union MO 1980 settlements on bank of Big Round Lake

ASU 289 All Soviet Union MO 1980 chord that maximise the area of ABCD

ASU 288 All Soviet Union MO 1980 diophantine x^2 + y^3 = z^4

ASU 287 All Soviet Union MO 1980 |AM| + |AP| = a, area of ABCD < a^2 /2

ASU 286 All Soviet Union MO 1980 35 containers, 3 metric tons each, 18 total

ASU 285 All Soviet Union MO 1980 strips inside a square

ASU 284 All Soviet Union MO 1980 all numbers from 19 to 80, divisible by 1980

1980 All Soviet Union Mathematical Olympiad

Subcontests

ASU 303 All Soviet Union MO 1980 rearranging 5 digits into a sequence, [0,1]

ASU 302 All Soviet Union MO 1980 tetrahedron , cos (AC,BD) &lt;= |CD|/|AB|

ASU 301 All Soviet Union MO 1980 [x^3/2]r +[y^3/2]=B has 1980 int. solutions

ASU 300 All Soviet Union MO 1980 integer set with min 1, max 100

ASU 299 All Soviet Union MO 1980 x&lt;1/3(p/4-sqrt(d^2-s/2)) in parallelepiped

ASU 298 All Soviet Union MO 1980 angle wanted, equilateral starting

ASU 297 All Soviet Union MO 1980 n_{k+1} = n_k + P(n_k), product of digits

ASU 296 All Soviet Union MO 1980 epidemic influenza and elfs

ASU 295 All Soviet Union MO 1980 red squares in infinite sheet of paper

ASU 294 All Soviet Union MO 1980 n+S(n)=1980, sum of digits

ASU 293 All Soviet Union MO 1980 1980 vectors in plane

ASU 292 All Soviet Union MO 1980 trigonometric system of equation

ASU 291 All Soviet Union MO 1980 dividible by 37

ASU 290 All Soviet Union MO 1980 settlements on bank of Big Round Lake

ASU 289 All Soviet Union MO 1980 chord that maximise the area of ABCD

ASU 288 All Soviet Union MO 1980 diophantine x^2 + y^3 = z^4

ASU 287 All Soviet Union MO 1980 |AM| + |AP| = a, area of ABCD &lt; a^2 /2

ASU 286 All Soviet Union MO 1980 35 containers, 3 metric tons each, 18 total

ASU 285 All Soviet Union MO 1980 strips inside a square

ASU 284 All Soviet Union MO 1980 all numbers from 19 to 80, divisible by 1980

ASU 302 All Soviet Union MO 1980 tetrahedron , cos (AC,BD) <= |CD|/|AB|

ASU 299 All Soviet Union MO 1980 x<1/3(p/4-sqrt(d^2-s/2)) in parallelepiped

ASU 287 All Soviet Union MO 1980 |AM| + |AP| = a, area of ABCD < a^2 /2