MathDB

1

ASU 124 All Soviet Union MO 1969 pentagon with equal sides

Given a pentagon with all equal sides.
a) Prove that there exist such a point on the maximal diagonal, that every side is seen from it inside a right angle. (side

AB

is seen from the point

C

inside an arbitrary angle that is greater or equal than

\angle ACB

)
b) Prove that the circles constructed on its sides as on the diameters cannot cover the pentagon entirely.

125

1

ASU 125 All Soviet Union MO 1969 x^3 + ?x^2 + ?x + ? = 0

Given an equation

x^3 + ?x^2 + ?x + ? = 0

First player substitutes an integer on the place of one of the interrogative marks, than the same do the second with one of the two remained marks, and, finally, the first puts the integer instead of the last mark. Explain how can the first provide the existence of three integer roots in the obtained equation. (The roots may coincide.)

128

1

ASU 128 All Soviet Union MO 1969 ... +a_n/(a_1+a_2)> n/4

Prove that for the arbitrary positive

a_1, a_2, ... , a_n

the following inequality is held

\frac{a_1}{a_2+a_3}+\frac{a_2}{a_3+a_4}+....+\frac{a_{n-1}}{a_n+a_1}+\frac{a_n}{a_1+a_2}>\frac{n}{4}

126

1

ASU 126 All Soviet Union MO 1969 20 football teams in championship

20

football teams participate in the championship. What minimal number of the games should be played to provide the property:
from the three arbitrary teams we can find at least on pair that have already met in the championship.

127

1

ASU 127All Soviet Union MO 1969 (n + 1)h_{n+1} - nh_n > R, apothem

Let

h_k

be an apothem of the regular

k

-gon inscribed into a circle with radius

R

. Prove that

(n + 1)h_{n+1} - nh_n > R

123

1

ASU 123 All Soviet Union MO 1969 airlines connecting cities

Every city in the certain state is connected by airlines with no more than with three other ones, but one can get from every city to every other city changing a plane once only or directly. What is the maximal possible number of the cities?

121

1

ASU 121 All Soviet Union MO 1969 n points in space angle > 120

Given

n

points in the three dimensional space such, that the arbitrary triangle with the vertices in three of those points contains an angle greater than

120

degrees. Prove that you can rearrange them to make a polyline (unclosed) with all the angles between the sequent links greater than

120

degrees.

122

1

ASU 122 All Soviet Union MO 1969 4 3-digit numbers, sum divisible by 3

Find four different three-digit decimal numbers starting with the same digit, such that their sum is divisible by three of them.

120

1

ASU 120 All Soviet Union MO 1969 sum of 1/(pq), with coprime

Given natural

n

. Consider all the fractions

1/(pq)

, where

p

and

q

are relatively prime,

0<p<q\le n , p+q>n

. Prove that the sum of all such a fractions equals to

1/2

.

119

1

ASU 119 All Soviet Union MO 1969 min a for ax^2 + bx + c

For what minimal natural

a

the polynomial

ax^2 + bx + c

with the integer

c

and

b

has two different positive roots both less than one.

118

1

ASU 118 All Soviet Union MO 1969 a+b<c+d, (a+b)(c+d)<ab+cd, (a+b)cd<ab(c+d)

Given positive numbers

a,b,c,d

. Prove that the set of inequalities

a+b<c+d

(a+b)(c+d)<ab+cd

(a+b)cd<ab(c+d)

contain at least one wrong.

117

1

ASU 117 All Soviet Union MO 1969 finite sequence of 0s and 1s

Given a finite sequence of zeros and ones, which has two properties:
a) if in some arbitrary place in the sequence we select five digits in a row and also select five digits in any other place in a row, then these fives will be different (they may overlap); b) if you add any digit to the right of the sequence, then property (a) will no longer hold true.
Prove that the first four digits of our sequence coincide with the last four

116

1

ASU 116 All Soviet Union MO 1969 a wolf and 4 dogs 1969 leningrad 7.4*

There is a wolf in the centre of a square field, and four dogs in the corners. The wolf can easily kill one dog, but two dogs can kill the wolf. The wolf can run all over the field, and the dogs -- along the fence (border) only. Prove that if the dog's speed is

1.5

times more than the wolf's, than the dogs can prevent the wolf escaping.

115

1

ASU 115 All Soviet Union MO 1969 equal perimeters inside trapezoid

The point

E

lies on the base

[AD]

of the trapezoid

ABCD

. The perimeters of the triangles

ABE, BCE

and

CDE

are equal. Prove that

|BC| = |AD|/2

1969 All Soviet Union Mathematical Olympiad

Subcontests

ASU 124 All Soviet Union MO 1969 pentagon with equal sides

ASU 125 All Soviet Union MO 1969 x^3 + ?x^2 + ?x + ? = 0

ASU 128 All Soviet Union MO 1969 ... +a_n/(a_1+a_2)> n/4

ASU 126 All Soviet Union MO 1969 20 football teams in championship

ASU 127All Soviet Union MO 1969 (n + 1)h_{n+1} - nh_n > R, apothem

ASU 123 All Soviet Union MO 1969 airlines connecting cities

ASU 121 All Soviet Union MO 1969 n points in space angle > 120

ASU 122 All Soviet Union MO 1969 4 3-digit numbers, sum divisible by 3

ASU 120 All Soviet Union MO 1969 sum of 1/(pq), with coprime

ASU 119 All Soviet Union MO 1969 min a for ax^2 + bx + c

ASU 118 All Soviet Union MO 1969 a+b<c+d, (a+b)(c+d)<ab+cd, (a+b)cd<ab(c+d)

ASU 117 All Soviet Union MO 1969 finite sequence of 0s and 1s

ASU 116 All Soviet Union MO 1969 a wolf and 4 dogs 1969 leningrad 7.4*

ASU 115 All Soviet Union MO 1969 equal perimeters inside trapezoid

1969 All Soviet Union Mathematical Olympiad

Subcontests

ASU 124 All Soviet Union MO 1969 pentagon with equal sides

ASU 125 All Soviet Union MO 1969 x^3 + ?x^2 + ?x + ? = 0

ASU 128 All Soviet Union MO 1969 ... +a_n/(a_1+a_2)&gt; n/4

ASU 126 All Soviet Union MO 1969 20 football teams in championship

ASU 127All Soviet Union MO 1969 (n + 1)h_{n+1} - nh_n &gt; R, apothem

ASU 123 All Soviet Union MO 1969 airlines connecting cities

ASU 121 All Soviet Union MO 1969 n points in space angle &gt; 120

ASU 122 All Soviet Union MO 1969 4 3-digit numbers, sum divisible by 3

ASU 120 All Soviet Union MO 1969 sum of 1/(pq), with coprime

ASU 119 All Soviet Union MO 1969 min a for ax^2 + bx + c

ASU 118 All Soviet Union MO 1969 a+b&lt;c+d, (a+b)(c+d)&lt;ab+cd, (a+b)cd&lt;ab(c+d)

ASU 117 All Soviet Union MO 1969 finite sequence of 0s and 1s

ASU 116 All Soviet Union MO 1969 a wolf and 4 dogs 1969 leningrad 7.4*

ASU 115 All Soviet Union MO 1969 equal perimeters inside trapezoid

ASU 128 All Soviet Union MO 1969 ... +a_n/(a_1+a_2)> n/4

ASU 127All Soviet Union MO 1969 (n + 1)h_{n+1} - nh_n > R, apothem

ASU 121 All Soviet Union MO 1969 n points in space angle > 120

ASU 118 All Soviet Union MO 1969 a+b<c+d, (a+b)(c+d)<ab+cd, (a+b)cd<ab(c+d)