MathDB

1

ASU 446 All Soviet Union MO 1987 tiling with Ls, of 3 adjacent unit squares

L

is an arrangement of

3

adjacent unit squares formed by deleting one unit square from a

2 \times 2

square.
a) How many

L

s can be placed on an

8 \times 8

board (with no interior points overlapping)?
b) Show that if any one square is deleted from a

1987 \times 1987

board, then the remaining squares can be covered with

L

s (with no interior points overlapping).

462

1

ASU 462 All Soviet Union MO 1987 (2n + 1)^n >= (2n)^n + (2n - 1)^n

Prove that for every natural

n

the following inequality is held:

(2n + 1)^n \ge (2n)^n + (2n - 1)^n

461

1

ASU 461 All Soviet Union MO 1987 faces of polyhedron are triangles, coloring

All the faces of a convex polyhedron are the triangles. Prove that it is possible to paint all its edges in red and blue colour in such a way, that it is possible to move from the arbitrary vertex to every vertex along the blue edges only and along the red edges only.

460

1

ASU 460 All Soviet Union MO 1987 rotate plot of a function, f(x)=x

The plot of the

y=f(x)

function, being rotated by the (right) angle around the

(0,0)

point is not changed.
a) Prove that the equation

f(x)=x

has the unique solution.
b) Give an example of such a function.

459

1

ASU 459 All Soviet Union MO 1987 T_0 set consists of all (2k)!

The

T_0

set consists of all the numbers, representable as

(2k)!, k = 0, 1, 2, ... , n, ...

. The

T_m

set is obtained from

T_{m-1}

by adding all the finite sums of different numbers, that belong to

T_{m-1}

. Prove that there is a natural number, that doesn't belong to

T_{1987}

.

458

1

ASU 458 All Soviet Union MO 1987 cutting a convex 5-gon along diagonals

The convex

n

-gon (

n\ge 5

) is cut along all its diagonals. Prove that there are at least a pair of parts with the different areas.

457

1

ASU 457 All Soviet Union MO 1987 lattice point, vectors, marked points

Some points with the integer coordinates are marked on the coordinate plane. Given a set of nonzero vectors. It is known, that if you apply the beginnings of those vectors to the arbitrary marked point, than there will be more marked ends of the vectors, than not marked. Prove that there is infinite number of marked points.

456

1

ASU 456 All Soviet Union MO 1987 9 or 10 of 33 knights in night guard

Every evening uncle Chernomor (see Pushkin's tales) appoints either

9

or

10

of his 33 "knights" in the "night guard". When it can happen, for the first time, that every knight has been on duty the same number of times?

455

1

ASU 455 All Soviet Union MO 1987 game with naturals <=p

Two players are writting in turn natural numbers not exceeding

p

. The rules forbid to write the divisors of the numbers already having been written. Those who cannot make his move looses.
a) Who, and how, can win if

p=10

?
b) Who wins if

p=1000

?

454

1

ASU 454 All Soviet Union MO 1987 PM=distance circumcenter from angle bis.

Vertex

B

of the

\angle ABC

lies out the circle, and the

[BA)

and

[BC)

beams intersect it. Point

K

belongs to the intersection of the

[BA)

beam and the circumference. Chord

KP

is orthogonal to the angle bisector of

\angle ABC

. Line

(KP)

intersects the beam

BC

in the point

M

. Prove that the segment

[PM]

is twice as long as the distance from the circle centre to the angle bisector of

\angle ABC

.

453

1

ASU 453 All Soviet Union MO 1987 numbers in 1987x1987 board

Each field of the

1987\times 1987

board is filled with numbers, which absolute value is not greater than one. The sum of all the numbers in every

2\times 2

square equals

0

. Prove that the sum of all the numbers is not greater than

1987

.

452

1

ASU 452 All Soviet Union MO 1987 a+A=b+B=c+C=k => aB+bC+cA <=k^2

The positive numbers

a,b,c,A,B,C

satisfy a condition

a + A = b + B = c + C = k

Prove that

aB + bC + cA \le k^2

451

1

ASU 451 All Soviet Union MO 1987 cos a, cos 2a, cos 4a, ... , cos((2^n)a)<0

Prove such

a

, that all the numbers

\cos a, \cos 2a, \cos 4a, ... , \cos (2^na)

are negative.

450

1

ASU 450 All Soviet Union MO 198equal angles, given and wanted in convex pentagon

Given a convex pentagon

ABCDE

with

\angle ABC= \angle ADE

and

\angle AEC= \angle ADB

. Prove that

\angle BAC = \angle DAE

.

449

1

ASU 449 All Soviet Union MO 1987 5 rel.primes such subset sum is composite

Find a set of five different relatively prime natural numbers such, that the sum of an arbitrary subset is a composite number.

448

1

ASU 448 All Soviet Union MO 1987 3 closed broken lines in plane, odd edges

Given two closed broken lines in the plane with odd numbers of edges. All the lines, containing those edges are different, and not a triple of them intersects in one point. Prove that it is possible to chose one edge from each line such, that the chosen edges will be the opposite sides of a convex quadrangle.

447

1

ASU 447 All Soviet Union MO 1987 3 lines // triangle sides, concyclic wanted

Three lines are drawn parallel to the sides of the triangles in the opposite to the vertex, not belonging to the side, part of the plane. The distance from each side to the corresponding line equals the length of the side. Prove that six intersection points of those lines with the continuations of the sides are situated on one circumference.

444

2

ASU 444 All Soviet Union MO 1987 Sea battle game, 7x7 table

The "Sea battle" game.
a) You are trying to find the

4

-field ship -- a rectangle

1x4

, situated on the

7x7

playing board. You are allowed to ask a question, whether it occupies the particular field or not. How many questions is it necessary to ask to find that ship surely?
b) The same question, but the ship is a connected (i.e. its fields have common sides) set of

4

fields.

ASU 445 All Soviet Union MO 1987 (n+2)/1^{1987} + 2^{1987} + ... + n^{1987}

Prove that

1^{1987} + 2^{1987} + ... + n^{1987}

is divisible by

n+2

.

443

1

ASU 443 All Soviet Union MO 1987 r. heptagon 1/A_1A_5+1/A_1A_3=1/A_1A_7

Given a regular heptagon

A_1...A_7

. Prove that

\frac{1}{|A_1A_5|} + \frac{1}{|A_1A_3| }= \frac{1}{|A_1A_7|}

.

442

1

ASU 442 All Soviet Union MO 1987 6 weighs to scale 1 to 63

It is known that, having

6

weighs, it is possible to balance the scales with loads, which weights are successing natural numbers from

1

to

63

. Find all such sets of weighs.

441

1

ASU 441 All Soviet Union MO 1987 tennis x_1^2+...+x_{10}^2=y_1^2+...+y_{10}^2

Ten sportsmen have taken part in a table-tennis tournament (each pair has met once only, no draws). Let

xi

be the number of

i

-th player victories,

yi

-- losses. Prove that

x_1^2 + ... + x_{10}^2 = y_1^2 + ... + y_{10}^2

1987 All Soviet Union Mathematical Olympiad

Subcontests

ASU 446 All Soviet Union MO 1987 tiling with Ls, of 3 adjacent unit squares

ASU 462 All Soviet Union MO 1987 (2n + 1)^n >= (2n)^n + (2n - 1)^n

ASU 461 All Soviet Union MO 1987 faces of polyhedron are triangles, coloring

ASU 460 All Soviet Union MO 1987 rotate plot of a function, f(x)=x

ASU 459 All Soviet Union MO 1987 T_0 set consists of all (2k)!

ASU 458 All Soviet Union MO 1987 cutting a convex 5-gon along diagonals

ASU 457 All Soviet Union MO 1987 lattice point, vectors, marked points

ASU 456 All Soviet Union MO 1987 9 or 10 of 33 knights in night guard

ASU 455 All Soviet Union MO 1987 game with naturals <=p

ASU 454 All Soviet Union MO 1987 PM=distance circumcenter from angle bis.

ASU 453 All Soviet Union MO 1987 numbers in 1987x1987 board

ASU 452 All Soviet Union MO 1987 a+A=b+B=c+C=k => aB+bC+cA <=k^2

ASU 451 All Soviet Union MO 1987 cos a, cos 2a, cos 4a, ... , cos((2^n)a)<0

ASU 450 All Soviet Union MO 198equal angles, given and wanted in convex pentagon

ASU 449 All Soviet Union MO 1987 5 rel.primes such subset sum is composite

ASU 448 All Soviet Union MO 1987 3 closed broken lines in plane, odd edges

ASU 447 All Soviet Union MO 1987 3 lines // triangle sides, concyclic wanted

ASU 444 All Soviet Union MO 1987 Sea battle game, 7x7 table

ASU 445 All Soviet Union MO 1987 (n+2)/1^{1987} + 2^{1987} + ... + n^{1987}

ASU 443 All Soviet Union MO 1987 r. heptagon 1/A_1A_5+1/A_1A_3=1/A_1A_7

ASU 442 All Soviet Union MO 1987 6 weighs to scale 1 to 63

ASU 441 All Soviet Union MO 1987 tennis x_1^2+...+x_{10}^2=y_1^2+...+y_{10}^2

1987 All Soviet Union Mathematical Olympiad

Subcontests

ASU 446 All Soviet Union MO 1987 tiling with Ls, of 3 adjacent unit squares

ASU 462 All Soviet Union MO 1987 (2n + 1)^n &gt;= (2n)^n + (2n - 1)^n

ASU 461 All Soviet Union MO 1987 faces of polyhedron are triangles, coloring

ASU 460 All Soviet Union MO 1987 rotate plot of a function, f(x)=x

ASU 459 All Soviet Union MO 1987 T_0 set consists of all (2k)!

ASU 458 All Soviet Union MO 1987 cutting a convex 5-gon along diagonals

ASU 457 All Soviet Union MO 1987 lattice point, vectors, marked points

ASU 456 All Soviet Union MO 1987 9 or 10 of 33 knights in night guard

ASU 455 All Soviet Union MO 1987 game with naturals &lt;=p

ASU 454 All Soviet Union MO 1987 PM=distance circumcenter from angle bis.

ASU 453 All Soviet Union MO 1987 numbers in 1987x1987 board

ASU 452 All Soviet Union MO 1987 a+A=b+B=c+C=k =&gt; aB+bC+cA &lt;=k^2

ASU 451 All Soviet Union MO 1987 cos a, cos 2a, cos 4a, ... , cos((2^n)a)&lt;0

ASU 450 All Soviet Union MO 198equal angles, given and wanted in convex pentagon

ASU 449 All Soviet Union MO 1987 5 rel.primes such subset sum is composite

ASU 448 All Soviet Union MO 1987 3 closed broken lines in plane, odd edges

ASU 447 All Soviet Union MO 1987 3 lines // triangle sides, concyclic wanted

ASU 444 All Soviet Union MO 1987 Sea battle game, 7x7 table

ASU 445 All Soviet Union MO 1987 (n+2)/1^{1987} + 2^{1987} + ... + n^{1987}

ASU 443 All Soviet Union MO 1987 r. heptagon 1/A_1A_5+1/A_1A_3=1/A_1A_7

ASU 442 All Soviet Union MO 1987 6 weighs to scale 1 to 63

ASU 441 All Soviet Union MO 1987 tennis x_1^2+...+x_{10}^2=y_1^2+...+y_{10}^2

ASU 462 All Soviet Union MO 1987 (2n + 1)^n >= (2n)^n + (2n - 1)^n

ASU 455 All Soviet Union MO 1987 game with naturals <=p

ASU 452 All Soviet Union MO 1987 a+A=b+B=c+C=k => aB+bC+cA <=k^2

ASU 451 All Soviet Union MO 1987 cos a, cos 2a, cos 4a, ... , cos((2^n)a)<0