MathDB

1

ASU 199 All Soviet Union MO 1974 cats and rats game in 8x8 chessboard

Two are playing the game "cats and rats" on the chess-board

8\times 8

. The first has one piece -- a rat, the second -- several pieces -- cats. All the pieces have four available moves -- up, down, left, right -- to the neighbour field, but the rat can also escape from the board if it is on the boarder of the chess-board. If they appear on the same field -- the rat is eaten. The players move in turn, but the second can move all the cats in independent directions.
a) Let there be two cats. The rat is on the interior field. Is it possible to put the cats on such a fields on the border that they will be able to catch the rat?
b) Let there be three cats, but the rat moves twice during the first turn. Prove that the rat can escape.

200

1

ASU 200 All Soviet Union MO 1974 every couple of no.s not equal their mean

a) Prove that you can rearrange the numbers

1, 2, ... , 32

in such a way, that for every couple of numbers none of the numbers between them will equal their arithmetic mean.
b) Can you rearrange the numbers

1, 2, ... , 100

in such a way, that for every couple of numbers none of the numbers between them will equal their arithmetic mean?

204

1

ASU 204 All Soviet Union MO 1974 min common area of two triangles

Given a triangle

ABC

with the are

1

. Let

A',B'

and

C'

are the midpoints of the sides

[BC], [CA]

and

[AB]

respectively. What is the minimal possible area of the common part of two triangles

A'B'C'

and

KLM

, if the points

K,L

and

M

are lying on the segments

[AB'], [CA']

and

[BC']

respectively?

203

1

ASU 203 All Soviet Union MO 1974 f(x_1+x_2)\ge f(x_1)+f(x_2)

Given a function

f(x)

on the segment

0\le x\le 1

. For all

x, f(x)\ge 0, f(1)=1

. For all the couples of

(x_1,x_2)

such, that all the arguments are in the segment

f(x_1+x_2)\ge f(x_1)+f(x_2).

a) Prove that for all

x

holds

f(x) \le 2x

.
b) Is the inequality

f(x) \le 1.9x

valid?

202

1

ASU 202 All Soviet Union MO 1974 convex polygon inside a triangle of area 4

Given a convex polygon. You can put no triangle with area

1

inside it. Prove that you can put the polygon inside a triangle with the area

4

.

201

1

ASU 201 All Soviet Union MO 1974 no= mean of numbers arranging digits

Find all the three-digit numbers such that it equals to the arithmetic mean of the six numbers obtained by rearranging its digits.

198

1

ASU 198 All Soviet Union MO 1974 equal segments, right isosceles

Given points

D

and

E

on the legs

[CA]

and

[CB]

, respectively, of the isosceles right triangle.

|CD| = |CE|

. The extensions of the perpendiculars from

D

and

C

to the line

AE

cross the hypotenuse

AB

in the points

K

and

L

. Prove that

|KL| = |LB|

196

1

ASU 196 All Soviet Union MO 1974 connecting red, blue points

Given some red and blue points. Some of them are connected by the segments. Let us call "exclusive" the point, if its colour differs from the colour of more than half of the connected points. Every move one arbitrary "exclusive" point is repainted to the other colour. Prove that after the finite number of moves there will remain no "exclusive" points.

197

1

ASU 197 All Soviet Union MO 1974 n^n has k digits and k^k has n digits

Find all the natural

n

and

k

such that

n^n

has

k

digits and

k^k

has

n

digits.

195

1

ASU 195 All Soviet Union MO 1974 right angle inside a square

Given a square

ABCD

. Points

P

and

Q

are in the sides

[AB]

and

[BC]

respectively.

|BP|=|BQ|

. Let

H

be the foot of the perpendicular from the point

B

to the segment

[PC]

. Prove that the

\angle DHQ =90^o

.

194

1

ASU 194 All Soviet Union MO 74 |ax+by+cz|+|bx+cy+az|+|cx+ay+bz|=|x|+|y|+|z|

Find all the real

a,b,c

such that the equality

|ax+by+cz| + |bx+cy+az| + |cx+ay+bz| = |x|+|y|+|z|

is valid for all the real

x,y,z

.

193

1

ASU 193 All Soviet Union MO 1974 n vectors in space, sum less than 2

Given

n

vectors of unit length in the plane. The length of their total sum is less than one. Prove that you can rearrange them to provide the property:
for every

k, k\le n

, the length of the sum of the first

k

vectors is less than

2

.

192

1

ASU 192 All Soviet Union MO 1974 min side of trapezoid, tangent circles

Given two circles with the radiuses

R

and

r

, touching each other from the outer side. Consider all the trapezoids, such that its lateral sides touch both circles, and its bases touch different circles. Find the shortest possible lateral side.

191

1

ASU 191 All Soviet Union MO 1974 diagonal of convex hexagon longer than

a) Each of the side of the convex hexagon is longer than

1

. Does it necessary have a diagonal longer than

2

?
b) Each of the main diagonals of the convex hexagon is longer than

2

. Does it necessary have a side longer than

1

?

190

1

ASU 190 All Soviet Union MO 1974 min of no in form 36^k - 5^l

Among all the numbers representable as

36^k - 5^l

(

k

and

l

are natural numbers) find the smallest. Prove that it is really the smallest.

189

1

ASU 189 All Soviet Union MO 1974 triples of cards with -1 or +1, product

Given some cards with either "

-1

" or "

+1

" written on the opposite side. You are allowed to choose a triple of cards and ask about the product of the three numbers on the cards. What is the minimal number of questions allowing to determine all the numbers on the cards ...
a) for

30

cards,
b) for

31

cards,
c) for

32

cards.
(You should prove, that you cannot manage with less questions.)
d) Fifty above mentioned cards are lying along the circumference. You are allowed to ask about the product of three consecutive numbers only. You need to determine the product af all the

50

numbers. What is the minimal number of questions allowing to determine it?

1974 All Soviet Union Mathematical Olympiad

Subcontests

ASU 199 All Soviet Union MO 1974 cats and rats game in 8x8 chessboard

ASU 200 All Soviet Union MO 1974 every couple of no.s not equal their mean

ASU 204 All Soviet Union MO 1974 min common area of two triangles

ASU 203 All Soviet Union MO 1974 f(x_1+x_2)\ge f(x_1)+f(x_2)

ASU 202 All Soviet Union MO 1974 convex polygon inside a triangle of area 4

ASU 201 All Soviet Union MO 1974 no= mean of numbers arranging digits

ASU 198 All Soviet Union MO 1974 equal segments, right isosceles

ASU 196 All Soviet Union MO 1974 connecting red, blue points

ASU 197 All Soviet Union MO 1974 n^n has k digits and k^k has n digits

ASU 195 All Soviet Union MO 1974 right angle inside a square

ASU 194 All Soviet Union MO 74 |ax+by+cz|+|bx+cy+az|+|cx+ay+bz|=|x|+|y|+|z|

ASU 193 All Soviet Union MO 1974 n vectors in space, sum less than 2

ASU 192 All Soviet Union MO 1974 min side of trapezoid, tangent circles

ASU 191 All Soviet Union MO 1974 diagonal of convex hexagon longer than

ASU 190 All Soviet Union MO 1974 min of no in form 36^k - 5^l

ASU 189 All Soviet Union MO 1974 triples of cards with -1 or +1, product