MathDB

1

ASU 105 All Soviet Union MO 1968 signs +, or - in a square table

4\times 4

are filled with the "+" or "-" signs. You are allowed to change the signs simultaneously in the whole row, column, or diagonal to the opposite sign. That means, for example, that You can change the sign in the corner square, because it makes a diagonal itself. Prove that you will never manage to obtain a table containing pluses only.
b) The fields of the square table

8\times 8

are filled with the "+" or signs except one non-corner field with "-". You are allowed to change the signs simultaneously in the whole row, column, or diagonal to the opposite sign. That means, for example, that You can change the sign in the corner field, because it makes a diagonal itself. Prove that you will never manage to obtain a table containing pluses only.

108

1

ASU 108 All Soviet Union MO 1968 9 referees on skating championship

Each of the

9

referees on the figure skating championship estimates the program of

20

sportsmen by assigning him a place (from

1

to

20

). The winner is determined by adding those numbers. (The less is the sum - the higher is the final place). It was found, that for the each sportsman, the difference of the places, received from the different referees was not greater than

3

. What can be the maximal sum for the winner?

109

1

ASU 109 All Soviet Union MO 1968 1, 1/2, ... , 1/n

Two finite sequences

a_1,a_2,...,a_n,b_1,b_2,...,b_n

are just rearranged sequence

1, 1/2, ... , 1/n

with

a_1+b_1\ge a_2+b_2\ge...\ge a_n+b_n.

Prove that

a_m+a_n\ge 4/m

for every

m

(

1\le m\le n

) .

104

1

ASU 104 All Soviet Union MO 1968 a tetrahedron and 4 spheres

Three spheres are constructed so that the edges

[AB], [BC], [AD]

of the tetrahedron

ABCD

are their respective diameters. Prove that the spheres cover all the tetrahedron.

101

1

ASU 101 All Soviet Union MO 1968 parallel and equal products wanted

Given two acute-angled triangles

ABC

and

A'B'C'

with the points

O

and

O'

inside. Three pairs of the perpendiculars are drawn:

[OA_1]

to the side

[BC]

,

[O'A'_1]

to the side

[B'C']

,

[OB_1]

to the side

[AC]

,

[O'B'_1]

to the side

[A'C']

,

[OC_1]

to the side

[AB]

,

[O'C'_1]

to the side

[A'B']

; It is known that

[OA_1] \parallel [O'A'], [OB_1] \parallel [O'B'], [OC_1] \parallel [O'C']

and

|OA_1|\cdot|O'A'| = |OB_1|\cdot |O'B'| = |OC_1|\cdot |O'C'|

Prove that

[O'A'_1] \parallel [OA], [O'B'_1] \parallel[OB], [O'C'_1] \parallel[OC]

and

|O'A'_1|\cdot|OA| = |O'B'_1|\cdot|OB| = |O'C'_1|\cdot|OC|

098

1

ASU 098 All Soviet Union MO 1968 2/(x^2-1)+..+20/(x^2-100)= ..

Prove the equality

\frac{2}{x^2-1}+\frac{4}{x^2-4} +\frac{6}{x^2-9}+...+\frac{20}{x^2-100} =\frac{11}{(x-1)(x+10)}+\frac{11}{(x-2)(x+9)}+...+\frac{11}{(x-10)(x+1)}

111

1

ASU 111 All Soviet Union MO 1968 policemen and crossroads

The city is a rectangle divided onto squares by

m

streets coming from the West to the East and

n

streets coming from the North to the South. There are militioners (policemen) on the streets but not on the crossroads. They watch the certain automobile, moving along the closed route, marking the time and the direction of its movement. Its trace is not known in advance, but they know, that it will not pass over the same segment of the way twice. What is the minimal number of the militioners providing the unique determination of the route according to their reports?

113

1

ASU 113 All Soviet Union MO 1968 |a_n|=|a_{n-1}+1|

The sequence

a_1,a_2,...,a_n

satisfies the following conditions:

a_1=0, |a_2|=|a_1+1|, ..., |a_n|=|a_{n-1}+1|.

Prove that

(a_1+a_2+...+a_n)/n \ge -1/2

110

1

ASU 110 All Soviet Union MO 1968 scales and pupils

There is scales on the teacher's table. There is a set of weighs on the scales, and there are some pupils' names (may be more than one) on the every weigh. A pupil entering the classroom moves all the weight with his name to another side of the scales. Prove that you can let in such a subset of the pupils, that the scales will change its position.

112

1

ASU 112 All Soviet Union MO 1968 incircle related

The circle inscribed in the triangle

ABC

touches the side

[AC]

in the point

K

. Prove that the line connecting the midpoint of the side

[AC]

with the centre of the circle halves the segment

[BK]

.

114

1

ASU 114 All Soviet Union MO 1968 rational sides and diagonals

Given a quadrangle

ABCD

. The lengths of all its sides and diagonals are the rational numbers. Let

O

be the point of its diagonals intersection. Prove that

|AO|

- the length of the

[AO]

segment is also rational.

107

1

ASU 107 All Soviet Union MO 1968 x^2 + x + 1 = py

Prove that the equation

x^2 + x + 1 = py

has solution

(x,y)

for the infinite number of simple

p

.

106

1

ASU 106 All Soviet Union MO 1968 4 equal incircles by medians

Medians divide the triangle onto

6

smaller ones.

4

of the circles inscribed in those small ones are equal. Prove that the triangle is equilateral.

102

1

ASU 102 All Soviet Union MO 1968 no as sum of divisors <= n!

Prove that you can represent an arbitrary number not exceeding

n!

as a sum of

k

different numbers (

k\le n

) that are divisors of

n!

.

103

1

ASU 103 All Soviet Union MO 1968 decagon

Given a triangle

ABC

, point

D

on

[AB], E

on

[AC]

,

|AD| = |DE| = |AC| , |BD| = |AE| , DE

is parallel to

BC

. Prove that the length

|BD|

equals to the side of a regular decagon inscribed in a circle with the radius

R=|AC|

.

100

1

ASU 100 All Soviet Union MO 1968 a_{n+1}=a_n+1/a_n

The sequence

a_1,a_2,a_3,...

, is constructed according to the rule

a_1=1, a_2=a_1+1/a_1, ... , a_{n+1}=a_n+1/a_n, ...

Prove that

a_{100} > 14

.

099

1

ASU 099 All Soviet Union MO 1968 difference of min and max diagonal

The difference between the maximal and the minimal diagonals of the regular

n

-gon equals to its side (

n > 5

). Find

n

.

097

1

ASU 097 All Soviet Union MO 1968 students and languages

Some students on the faculty speak several languages and some - Russian only.

50

of them know English,

50

-- French and

50

-- Spanish. Prove that it is possible to divide them onto

5

groups, not necessary equal, to get

10

of them knowing English,

10

-- French and

10

-- Spanish in each group.

096

1

ASU 096 All Soviet Union MO 1968 squares and circle

The circumference with the radius

100

cm is drawn on the cross-lined paper with the side of the squares

1

cm. It neither comes through the vertices of the squares, nor touches the lines. How many squares can it pass through?

095

1

ASU 095 All Soviet Union MO 1968 compare 31^{11} with 17^{14}

What is greater,

31^{11}

or

17^{14}

?

094

1

ASU 094 All Soviet Union MO 1968 octagon with equal angles and integer sides

Given an octagon with the equal angles. The lengths of all the sides are integers. Prove that the opposite sides are equal in pairs.
alternate wording Consider an octagon with equal angles and with rational sides. Prove that it has a center of symmetry.

1968 All Soviet Union Mathematical Olympiad

Subcontests

ASU 105 All Soviet Union MO 1968 signs +, or - in a square table

ASU 108 All Soviet Union MO 1968 9 referees on skating championship

ASU 109 All Soviet Union MO 1968 1, 1/2, ... , 1/n

ASU 104 All Soviet Union MO 1968 a tetrahedron and 4 spheres

ASU 101 All Soviet Union MO 1968 parallel and equal products wanted

ASU 098 All Soviet Union MO 1968 2/(x^2-1)+..+20/(x^2-100)= ..

ASU 111 All Soviet Union MO 1968 policemen and crossroads

ASU 113 All Soviet Union MO 1968 |a_n|=|a_{n-1}+1|

ASU 110 All Soviet Union MO 1968 scales and pupils

ASU 112 All Soviet Union MO 1968 incircle related

ASU 114 All Soviet Union MO 1968 rational sides and diagonals

ASU 107 All Soviet Union MO 1968 x^2 + x + 1 = py

ASU 106 All Soviet Union MO 1968 4 equal incircles by medians

ASU 102 All Soviet Union MO 1968 no as sum of divisors <= n!

ASU 103 All Soviet Union MO 1968 decagon

ASU 100 All Soviet Union MO 1968 a_{n+1}=a_n+1/a_n

ASU 099 All Soviet Union MO 1968 difference of min and max diagonal

ASU 097 All Soviet Union MO 1968 students and languages

ASU 096 All Soviet Union MO 1968 squares and circle

ASU 095 All Soviet Union MO 1968 compare 31^{11} with 17^{14}

ASU 094 All Soviet Union MO 1968 octagon with equal angles and integer sides

1968 All Soviet Union Mathematical Olympiad

Subcontests

ASU 105 All Soviet Union MO 1968 signs +, or - in a square table

ASU 108 All Soviet Union MO 1968 9 referees on skating championship

ASU 109 All Soviet Union MO 1968 1, 1/2, ... , 1/n

ASU 104 All Soviet Union MO 1968 a tetrahedron and 4 spheres

ASU 101 All Soviet Union MO 1968 parallel and equal products wanted

ASU 098 All Soviet Union MO 1968 2/(x^2-1)+..+20/(x^2-100)= ..

ASU 111 All Soviet Union MO 1968 policemen and crossroads

ASU 113 All Soviet Union MO 1968 |a_n|=|a_{n-1}+1|

ASU 110 All Soviet Union MO 1968 scales and pupils

ASU 112 All Soviet Union MO 1968 incircle related

ASU 114 All Soviet Union MO 1968 rational sides and diagonals

ASU 107 All Soviet Union MO 1968 x^2 + x + 1 = py

ASU 106 All Soviet Union MO 1968 4 equal incircles by medians

ASU 102 All Soviet Union MO 1968 no as sum of divisors &lt;= n!

ASU 103 All Soviet Union MO 1968 decagon

ASU 100 All Soviet Union MO 1968 a_{n+1}=a_n+1/a_n

ASU 099 All Soviet Union MO 1968 difference of min and max diagonal

ASU 097 All Soviet Union MO 1968 students and languages

ASU 096 All Soviet Union MO 1968 squares and circle

ASU 095 All Soviet Union MO 1968 compare 31^{11} with 17^{14}

ASU 094 All Soviet Union MO 1968 octagon with equal angles and integer sides

ASU 102 All Soviet Union MO 1968 no as sum of divisors <= n!