MathDB

1

ASU 158 All Soviet Union MO 1971 switces, inputs and outputs

1, 2

and two outputs

1, 2

. It either connects

1

to

1

and

2

to

2

, or

1

to

2

and

2

to 1. If you have three inputs

1, 2, 3

and three outputs

1, 2, 3

, then you can use three switches, the first across

1

and

2

, then the second across

2

and

3

, and finally the third across

1

and

2

. It is easy to check that this allows the output to be any permutation of the inputs and that at least three switches are required to achieve this. What is the minimum number of switches required for

4

inputs, so that by suitably setting the switches the output can be any permutation of the inputs?

146

1

ASU 146 All Soviet Union MO 1971 game with hidden numbers, min questions

a) A game for two. The first player writes two rows of ten numbers each, the second under the first. He should provide the following property: if number

b

is written under

a

, and

d

-- under

c

, then

a + d = b + c

. The second player has to determine all the numbers. He is allowed to ask the questions like "What number is written in the

x

place in the

y

row?" What is the minimal number of the questions asked by the second player before he founds out all the numbers?
b) There was a table

m\times n

on the blackboard with the property: if You chose two rows and two columns, then the sum of the numbers in the two opposite vertices of the rectangles formed by those lines equals the sum of the numbers in two another vertices. Some of the numbers are cleaned but it is still possible to restore all the table. What is the minimal possible number of the remaining numbers?

157

1

ASU 157 All Soviet Union MO 1971 f_a(x,y) = x^2 + axy + y^2

a) Consider the function

f(x,y) = x^2 + xy + y^2

Prove that for the every point

(x,y)

there exist such integers

(m,n)

, that

f((x-m),(y-n)) \le 1/2

b) Let us denote with

g(x,y)

the least possible value of the

f((x-m),(y-n))

for all the integers

m,n

. The statement a) was equal to the fact

g(x,y) \le 1/2

. Prove that in fact,

g(x,y) \le 1/3

Find all the points

(x,y)

, where

g(x,y)=1/3

.
c) Consider function

f_a(x,y) = x^2 + axy + y^2 \,\,\, (0 \le a \le 2)

Find any

c

such that

g_a(x,y) \le c

. Try to obtain the closest estimation.

156

1

ASU 156 All Soviet Union MO 1971 n^3 cubes inside a cube of length n

A cube with the edge of length

n

is divided onto

n^3

unit ones. Let us choose some of them and draw three lines parallel to the edges through their centres. What is the least possible number of the chosen small cubes necessary to make those lines cross all the smaller cubes?
a) Find the answer for the small

n

(

n = 2,3,4

).
b) Try to find the answer for

n = 10

.
c) If You can not solve the general problem, try to estimate that value from the upper and lower side.
d) Note, that You can reformulate the problem in such a way:
Consider all the triples

(x_1,x_2,x_3)

, where

x_i

can be one of the integers

1,2,...,n

. What is the minimal number of the triples necessary to provide the property:
for each of the triples there exist the chosen one, that differs only in one coordinate.
Try to find the answer for the situation with more than three coordinates, for example, with four.

155

1

ASU 155 All Soviet Union MO 1971 black unit squares in an inifinite board

N

unit squares on the infinite sheet of cross-lined paper are painted with black colour. Prove that you can cut out the finite number of square pieces and satisfy two conditions all the black squares are contained in those pieces the area of black squares is not less than

1/5

and not greater than

4/5

of every piece area.

154

1

ASU 154 All Soviet Union MO 1971 signs +,- on vertices of a dodecagon

a) The vertex

A_1

of the regular

12

-gon (dodecagon)

A_1A_2...A_{12}

is marked with "

-

" and all the rest

--

with "

+

". You are allowed to change the sign simultaneously in the

6

vertices in succession. Prove that is impossible to obtain dodecagon with

A_2

marked with "

-

" and the rest of the vertices

--

with "

+

".
b) Prove the same statement if it is allowed to change the signs not in six, but in four vertices in succession.
c) Prove the same statement if it is allowed to change the signs in three vertices in succession.

153

1

ASU 153 All Soviet Union MO 1971 two of 25 such other 23 different from +,-

Given

25

different positive numbers. Prove that you can choose two of them such, that none of the other numbers equals neither to the sum nor to the difference between the chosen numbers.

152

1

ASU 152 All Soviet Union MO 19 equal perimeter + equal areas through incenter

a) Prove that the line dividing the triangle onto two polygons with equal perimeters and equal areas passes through the centre of the inscribed circle.
b) Prove the same statement for the arbitrary tangential polygon.
c) Prove that all the lines halving its perimeter and area simultaneously, intersect in one point.

151

1

ASU 151 All Soviet Union MO 1971 (a-d)(b-c)<0 for some numbers in ring

Some numbers are written along the ring. If inequality

(a-d)(b-c) < 0

is held for the four arbitrary numbers in sequence

a,b,c,d

, you have to change the numbers

b

and

c

places. Prove that you will have to do this operation finite number of times.

150

1

ASU 150 All Soviet Union MO 1971 projection on 2 planes are circles, same r

The projections of the body on two planes are circles. Prove that they have the same radius.

149

1

ASU 149 All Soviet Union MO 1971 trinomials with real roots

Prove that if the numbers

p_1, p_2, q_1, q_2

satisfy the condition

(q_1 - q_2)^2 + (p_1 - p_2)(p_1q_2 -p_2q_1)<0

then the square polynomials

x^2 + p_1x + q_1

and

x^2 + p_2x + q_2

have real roots, and between the roots of each there is a root of another one.

148

1

ASU 148 All Soviet Union MO 1971 3 containers with integer volumes of water

The volumes of the water containing in each of three big enough containers are integers. You are allowed only to relocate some times from one container to another the same volume of the water, that the destination already contains. Prove that you are able to discharge one of the containers.

147

1

ASU 147 All Soviet Union MO 1971 circles inside a square

Given an unit square and some circles inside. Radius of each circle is less than

0.001

, and there is no couple of points belonging to the different circles with the distance between them

0.001

exactly. Prove that the area, covered by the circles is not greater than

0.34

.

145

1

ASU 145 All Soviet Union MO 1971 n parallelograms on a convex n-gon

a) Given a triangle

A_1A_2A_3

and the points

B_1

and

D_2

on the side

[A_1A_2]

,

B_2

and

D_3

on the side

[A_2A3]

,

B_3

and

D_1

on the side

[A_3A_1]

. If you construct parallelograms

A_1B_1C_1D_1

,

A_2B_2C_2D_2

and

A_3B_3C_3D_3

, the lines

(A_1C_1)

,

(A_2C_2)

and

(A_3C_3)

, will cross in one point

O

. Prove that if

|A_1B_1| = |A_2D_2| \,\,\, and \,\,\, |A_2B_2| = |A_3D_3|

then

|A_3B_3| = |A_1D_1|

b) Given a convex polygon

A_1A_2 ... A_n

and the points

B_1

and

D_2

on the side

[A_1A_2]

,

B_2

and

D_3

on the side

[A_2A_3]

, ...

B_n

and

D_1

on the side

[A_nA_1]

. If you construct parallelograms

A_1B_1C_1D_1

,

A_2B_2C_2D_2

,

...

,

A_nB_nC_nD_n

, the lines

(A_1C_1)

,

(A_2C_2)

,

...

,

(A_nC_n)

, will cross in one point

O

. Prove that

|A_1B_1| \cdot |A_2B_2|\cdot ... \cdot |A_nB_n| = |A_1D_1|\cdot |A_2D_2|\cdot ...\cdot |A_nD_n|

144

1

ASU 144 All Soviet Union MO 1971 digits 1 and 2 in decimal such 2^n divides

Prove that for every natural

n

there exists a number, containing only digits "

1

" and "

2

" in its decimal notation, that is divisible by

2^n

(

n

-th power of two ).

1971 All Soviet Union Mathematical Olympiad

Subcontests

ASU 158 All Soviet Union MO 1971 switces, inputs and outputs

ASU 146 All Soviet Union MO 1971 game with hidden numbers, min questions

ASU 157 All Soviet Union MO 1971 f_a(x,y) = x^2 + axy + y^2

ASU 156 All Soviet Union MO 1971 n^3 cubes inside a cube of length n

ASU 155 All Soviet Union MO 1971 black unit squares in an inifinite board

ASU 154 All Soviet Union MO 1971 signs +,- on vertices of a dodecagon

ASU 153 All Soviet Union MO 1971 two of 25 such other 23 different from +,-

ASU 152 All Soviet Union MO 19 equal perimeter + equal areas through incenter

ASU 151 All Soviet Union MO 1971 (a-d)(b-c)<0 for some numbers in ring

ASU 150 All Soviet Union MO 1971 projection on 2 planes are circles, same r

ASU 149 All Soviet Union MO 1971 trinomials with real roots

ASU 148 All Soviet Union MO 1971 3 containers with integer volumes of water

ASU 147 All Soviet Union MO 1971 circles inside a square

ASU 145 All Soviet Union MO 1971 n parallelograms on a convex n-gon

ASU 144 All Soviet Union MO 1971 digits 1 and 2 in decimal such 2^n divides

1971 All Soviet Union Mathematical Olympiad

Subcontests

ASU 158 All Soviet Union MO 1971 switces, inputs and outputs

ASU 146 All Soviet Union MO 1971 game with hidden numbers, min questions

ASU 157 All Soviet Union MO 1971 f_a(x,y) = x^2 + axy + y^2

ASU 156 All Soviet Union MO 1971 n^3 cubes inside a cube of length n

ASU 155 All Soviet Union MO 1971 black unit squares in an inifinite board

ASU 154 All Soviet Union MO 1971 signs +,- on vertices of a dodecagon

ASU 153 All Soviet Union MO 1971 two of 25 such other 23 different from +,-

ASU 152 All Soviet Union MO 19 equal perimeter + equal areas through incenter

ASU 151 All Soviet Union MO 1971 (a-d)(b-c)&lt;0 for some numbers in ring

ASU 150 All Soviet Union MO 1971 projection on 2 planes are circles, same r

ASU 149 All Soviet Union MO 1971 trinomials with real roots

ASU 148 All Soviet Union MO 1971 3 containers with integer volumes of water

ASU 147 All Soviet Union MO 1971 circles inside a square

ASU 145 All Soviet Union MO 1971 n parallelograms on a convex n-gon

ASU 144 All Soviet Union MO 1971 digits 1 and 2 in decimal such 2^n divides

ASU 151 All Soviet Union MO 1971 (a-d)(b-c)<0 for some numbers in ring