MathDB

1

ASU 506 All Soviet Union MO 1989 equal altitudes of 2 walkers on mountains

Two walkers are at the same altitude in a range of mountains. The path joining them is piecewise linear with all its vertices above the two walkers. Can they each walk along the path until they have changed places, so that at all times their altitudes are equal?

505

1

ASU 505 All Soviet Union MO 1989 intersecting spheres, equal projections

S

and

S'

are two intersecting spheres. The line

BXB'

is parallel to the line of centers, where

B

is a point on

S, B'

is a point on

S'

and

X

lies on both spheres.

A

is another point on

S

, and

A'

is another point on S' such that the line

AA'

has a point on both spheres. Show that the segments

AB

and

A'B'

have equal projections on the line

AA'

.

510

1

ASU 510 All Soviet Union MO 1989 convex polygon area < \pi /4

A convex polygon is such that any segment dividing the polygon into two parts of equal area which has at least one end at a vertex has length

< 1

. Show that the area of the polygon is

< \pi /4

.

509

1

ASU 509 All Soviet Union MO 1989 f(n+1) = f( f(n) ) + f( f(n+2) )

N

is the set of positive integers. Does there exist a function

f: N \to N

such that

f(n+1) = f( f(n) ) + f( f(n+2) )

for all

n

?

508

1

ASU 508 All Soviet Union MO 1989 placing arrows on each vertex of a polyhedron

A polyhedron has an even number of edges. Show that we can place an arrow on each edge so that each vertex has an even number of arrows pointing towards it (on adjacent edges).

507

1

ASU 507 All Soviet Union MO 1989 min of (x+y)(y+z) when (x+y+z)xyz=1, x,y,z>0

Find the least possible value of

(x + y)(y + z)

for positive reals satisfying

(x + y + z) xyz = 1

.

504

1

ASU 504 All Soviet Union MO 1989 circumcenter of AEF lies on bisector of EDF

ABC

is a triangle. Points

D, E, F

are chosen on

BC, CA, AB

such that

B

is equidistant from

D

and

F

, and

C

is equidistant from

D

and

E

. Show that the circumcenter of

AEF

lies on the bisector of

EDF

.

503

1

ASU 503 All Soviet Union MO 1989 smallest n, exists m, [10^n/m]=1989

Find the smallest positive integer

n

for which we can find an integer

m

such that

\left[\frac{10^n}{m}\right] = 1989

.

502

1

ASU 502 All Soviet Union MO 1989 lattice polygon, disected in 1x2 in n ways

Show that for each integer

n > 0

, there is a polygon with vertices at lattice points and all sides parallel to the axes, which can be dissected into

1 \times 2

(and / or

2 \times 1

) rectangles in exactly

n

ways.

501

1

ASU 501 All Soviet Union MO 1989 collinear midpoints wanted

ABCD

has

AB = CD

, but

AB

not parallel to

CD

, and

AD

parallel to

BC

. The triangle is

ABC

is rotated about

C

to

A'B'C

. Show that the midpoints of

BC, B'C

and

A'D

are collinear.

500

1

ASU 500 All Soviet Union MO 1989 insects jumps on a square ceiling

An insect is on a square ceiling side

1

. The insect can jump to the midpoint of the segment joining it to any of the four corners of the ceiling. Show that in

8

jumps it can get to within

1/100

of any chosen point on the ceiling

499

1

ASU 499 All Soviet Union MO 1989 2 reals with sum of nth powers irrational

Do there exist two reals whose sum is rational, but the sum of their

n

th powers is irrational for all

n > 1

? Do there exist two reals whose sum is irrational, but the sum of whose

n

th powers is rational for all

n > 1

?

498

1

ASU 498 All Soviet Union MO 1989 23x23 square tiled with 1x1, 2x2, 3x3

A

23 \times 23

square is tiled with

1 \times 1, 2 \times 2

and

3 \times 3

squares. What is the smallest possible number of

1 \times 1

squares?

497

1

ASU 497 All Soviet Union MO 1989 area XQYP/area ABCD < mn/(m^2 + mn + n^2)

ABCD

is a convex quadrilateral.

X

lies on the segment

AB

with

\frac{AX}{XB} = \frac{m}{n}

.

Y

lies on the segment

CD

with

\frac{CY}{YD} = \frac{m}{n}

.

AY

and

DX

intersect at

P

, and

BY

and

CX

intersect at

Q

. Show that

\frac{S_{XQYP}}{S_{ABCD}} < \frac{mn}{m^2 + mn + n^2}

.

496

1

ASU 496 All Soviet Union MO 1989 a^2+b^2 +c^2 +4abc < 1/2, perimeter 1 triangle

A triangle with perimeter

1

has side lengths

a, b, c

. Show that

a^2 + b^2 + c^2 + 4abc <\frac 12

.

495

1

ASU 495 All Soviet Union MO 1989 1998 normal coins, 1 heavy coin, 1 light

We are given

1998

normal coins,

1

heavy coin and

1

light coin, which all look the same. We wish to determine whether the average weight of the two abnormal coins is less than, equal to, or greater than the weight of a normal coin. Show how to do this using a balance

4

times or less.

494

1

ASU 494 All Soviet Union MO 1989 120 5-digit nos, permutations of 12345

Show that the

120

five digit numbers which are permutations of

12345

can be divided into two sets with each set having the same sum of squares.

493

1

ASU 493 All Soviet Union MO 1989 n bird-nests in a forest, change nests

One bird lives in each of

n

bird-nests in a forest. The birds change nests, so that after the change there is again one bird in each nest. Also for any birds

A, B, C, D

(not necessarily distinct), if the distance

AB < CD

before the change, then

AB > CD

after the change. Find all possible values of

n

.

492

1

ASU 492 All Soviet Union MO 1989 simila triangles wanted

ABC

is a triangle.

A' , B' , C'

are points on the segments

BC, CA, AB

respectively.

\angle B' A' C' = \angle A

,

\frac{AC'}{C'B} = \frac{BA' }{A' C} = \frac{CB'}{B'A}

. Show that

ABC

and

A'B'C'

are similar.

491

1

ASU 491 All Soviet Union MO 1989 8 pawns on a cheesbooard

Eight pawns are placed on a chessboard, so that there is one in each row and column. Show that an even number of the pawns are on black squares.

490

1

ASU 490 All Soviet Union MO 1989 12 positive divisors, d_m=(d_1+d_2+d_4)d_8

A positive integer

n

has exactly

12

positive divisors

1 = d_1 < d_2 < d_3 < ... < d_{12} = n

. Let

m = d_4 - 1

. We have

d_m = (d_1 + d_2 + d_4) d_8

. Find

n

.

489

1

ASU 489 All Soviet Union MO 1989 3 incircles have a common tangent

The incircle of

ABC

touches

AB

at

M

.

N

is any point on the segment

BC

. Show that the incircles of

AMN, BMN, ACN

have a common tangent.

488

1

ASU 488 All Soviet Union MO 1989 77 blocks each 3x3x1 assembled 7x9x11

Can

77

blocks each

3 \times 3 \times1

be assembled to form a

7 \times 9 \times 11

block?

487

1

ASU 487 All Soviet Union MO 1989 7 boys each went to a shop 3 times

7

boys each went to a shop

3

times. Each pair met at the shop. Show that

3

must have been in the shop at the same time.

1989 All Soviet Union Mathematical Olympiad

Subcontests

ASU 506 All Soviet Union MO 1989 equal altitudes of 2 walkers on mountains

ASU 505 All Soviet Union MO 1989 intersecting spheres, equal projections

ASU 510 All Soviet Union MO 1989 convex polygon area < \pi /4

ASU 509 All Soviet Union MO 1989 f(n+1) = f( f(n) ) + f( f(n+2) )

ASU 508 All Soviet Union MO 1989 placing arrows on each vertex of a polyhedron

ASU 507 All Soviet Union MO 1989 min of (x+y)(y+z) when (x+y+z)xyz=1, x,y,z>0

ASU 504 All Soviet Union MO 1989 circumcenter of AEF lies on bisector of EDF

ASU 503 All Soviet Union MO 1989 smallest n, exists m, [10^n/m]=1989

ASU 502 All Soviet Union MO 1989 lattice polygon, disected in 1x2 in n ways

ASU 501 All Soviet Union MO 1989 collinear midpoints wanted

ASU 500 All Soviet Union MO 1989 insects jumps on a square ceiling

ASU 499 All Soviet Union MO 1989 2 reals with sum of nth powers irrational

ASU 498 All Soviet Union MO 1989 23x23 square tiled with 1x1, 2x2, 3x3

ASU 497 All Soviet Union MO 1989 area XQYP/area ABCD < mn/(m^2 + mn + n^2)

ASU 496 All Soviet Union MO 1989 a^2+b^2 +c^2 +4abc < 1/2, perimeter 1 triangle

ASU 495 All Soviet Union MO 1989 1998 normal coins, 1 heavy coin, 1 light

ASU 494 All Soviet Union MO 1989 120 5-digit nos, permutations of 12345

ASU 493 All Soviet Union MO 1989 n bird-nests in a forest, change nests

ASU 492 All Soviet Union MO 1989 simila triangles wanted

ASU 491 All Soviet Union MO 1989 8 pawns on a cheesbooard

ASU 490 All Soviet Union MO 1989 12 positive divisors, d_m=(d_1+d_2+d_4)d_8

ASU 489 All Soviet Union MO 1989 3 incircles have a common tangent

ASU 488 All Soviet Union MO 1989 77 blocks each 3x3x1 assembled 7x9x11

ASU 487 All Soviet Union MO 1989 7 boys each went to a shop 3 times

1989 All Soviet Union Mathematical Olympiad

Subcontests

ASU 506 All Soviet Union MO 1989 equal altitudes of 2 walkers on mountains

ASU 505 All Soviet Union MO 1989 intersecting spheres, equal projections

ASU 510 All Soviet Union MO 1989 convex polygon area &lt; \pi /4

ASU 509 All Soviet Union MO 1989 f(n+1) = f( f(n) ) + f( f(n+2) )

ASU 508 All Soviet Union MO 1989 placing arrows on each vertex of a polyhedron

ASU 507 All Soviet Union MO 1989 min of (x+y)(y+z) when (x+y+z)xyz=1, x,y,z&gt;0

ASU 504 All Soviet Union MO 1989 circumcenter of AEF lies on bisector of EDF

ASU 503 All Soviet Union MO 1989 smallest n, exists m, [10^n/m]=1989

ASU 502 All Soviet Union MO 1989 lattice polygon, disected in 1x2 in n ways

ASU 501 All Soviet Union MO 1989 collinear midpoints wanted

ASU 500 All Soviet Union MO 1989 insects jumps on a square ceiling

ASU 499 All Soviet Union MO 1989 2 reals with sum of nth powers irrational

ASU 498 All Soviet Union MO 1989 23x23 square tiled with 1x1, 2x2, 3x3

ASU 497 All Soviet Union MO 1989 area XQYP/area ABCD &lt; mn/(m^2 + mn + n^2)

ASU 496 All Soviet Union MO 1989 a^2+b^2 +c^2 +4abc &lt; 1/2, perimeter 1 triangle

ASU 495 All Soviet Union MO 1989 1998 normal coins, 1 heavy coin, 1 light

ASU 494 All Soviet Union MO 1989 120 5-digit nos, permutations of 12345

ASU 493 All Soviet Union MO 1989 n bird-nests in a forest, change nests

ASU 492 All Soviet Union MO 1989 simila triangles wanted

ASU 491 All Soviet Union MO 1989 8 pawns on a cheesbooard

ASU 490 All Soviet Union MO 1989 12 positive divisors, d_m=(d_1+d_2+d_4)d_8

ASU 489 All Soviet Union MO 1989 3 incircles have a common tangent

ASU 488 All Soviet Union MO 1989 77 blocks each 3x3x1 assembled 7x9x11

ASU 487 All Soviet Union MO 1989 7 boys each went to a shop 3 times

ASU 510 All Soviet Union MO 1989 convex polygon area < \pi /4

ASU 507 All Soviet Union MO 1989 min of (x+y)(y+z) when (x+y+z)xyz=1, x,y,z>0

ASU 497 All Soviet Union MO 1989 area XQYP/area ABCD < mn/(m^2 + mn + n^2)

ASU 496 All Soviet Union MO 1989 a^2+b^2 +c^2 +4abc < 1/2, perimeter 1 triangle