MathDB

1

ASU 071 All Russian MO 1965 spaceship finds the inhabitant on planet

On the surface of the planet lives one inhabitant, that can move with the speed not greater than

u

. A spaceship approaches to the planet with its speed

v

. Prove that if

v/u > 10

, the spaceship can find the inhabitant, even it is trying to hide.

070

1

ASU 070 All Russian MO 1965 sum of lengths of polyhedron edges more than

Prove that the sum of the lengths of the polyhedron edges exceeds its tripled diameter (distance between two farest vertices).

069

1

ASU 069 All Russian MO 1965 rocker hits the airplane-spy

A spy airplane flies on the circle with the centre

A

and radius

10

km. Its speed is

1000

km/h. At a certain moment, a rocket , that has same speed with the airplane, is launched from point

A

and moves along on the straight line connecting the airplane and point

A

.How long after launch will the rocket hit the plane?

068

1

ASU 068 All Russian MO 1965 n = px + qy, relative primes p,q

Given two relatively prime numbers

p>0

and

q>0

. An integer

n

is called "good" if we can represent it as

n = px + qy

with nonnegative integers

x

and

y

, and "bad" in the opposite case.
a) Prove that there exist integer

c

such that in a pair

\{n, c-n\}

always one is "good" and one is "bad".
b) How many there exist "bad" numbers?

067

1

ASU 067 All Russian MO 1965 members of subcommittees

a) A certain committee has gathered

40

times. There were

10

members on every meeting. Not a single couple has met on the meetings twice. Prove that there were no less then

60

members in the committee.
b) Prove that you can not construct more then

30

subcommittees of

5

members from the committee of

25

members, with no couple of subcommittees having more than one common member.

066

1

ASU 066 All Russian MO 1965 tourist in Moscow

The tourist has come to the Moscow by train. All-day-long he wandered randomly through the streets. Than he had a supper in the cafe on the square and decided to return to the station only through the streets that he has passed an odd number of times. Prove that he is always able to do that.

065

1

ASU 065 All Russian MO 1965 quasi-rounding n numbers

Quasi-rounding is a substitution one of the two closest integers instead of the given number. Given

n

numbers. Prove that you can quasi-round them in such a way, that a sum of every subset of quasi-rounded numbers will deviate from the sum of the same subset of initial numbers not greater than

(n+1)/4

.

064

1

ASU 064 All Russian MO 1965, 1965 points inside the unit square

Is it possible to place

1965

points in a square with side

1

so that any rectangle of area

1/200

with sides parallel to the sides of the square contains at least one of these points inside?

063

1

ASU 063 All Russian MO 1965 x_{i,j}+x_{j,k}+x_{k,i}=0

Given

n^2

numbers

x_{i,j}

(

i,j=1,2,...,n

) satisfying the system of

n^3

equations

x_{i,j}+x_{j,k}+x_{k,i}=0 \,\,\, (i,j,k = 1,...,n)

Prove that there exist such numbers

a_1,a_2,...,a_n

, that

x_{i,j}=a_i-a_j

for all

i,j=1,...n

.

062

1

ASU 062 All Russian MO 1965 max length of segment, incircle related

What is the maximal possible length of the segment, being cut out by the sides of the triangle on the tangent to the inscribed circle, being drawn parallel to the base, if the triangle's perimeter equals

2p

?

061

1

ASU 061 All Russian MO 1965 100 men help the police, 3 every evening

A society created in the help to the police contains

100

men exactly. Every evening

3

men are on duty. Prove that you can not organise duties in such a way, that every couple will meet on duty once exactly.

060

1

ASU 060 All Russian MO 1965 lighthouse enlights a ship

There is a lighthouse on a small island. Its lamp enlights a segment of a sea to the distance

a

. The light is turning uniformly, and the end of the segment moves with the speed

v

. Prove that a ship, whose speed doesn't exceed

v/8

cannot arrive to the island without being enlightened.

059

1

ASU 059 All Russian MO 1965 first 3 digits = last 3 digits, divisible by 13

A bus ticket is considered to be lucky if the sum of the first three digits equals to the sum of the last three (

6

digits in Russian buses). Prove that the sum of all the lucky numbers is divisible by

13

.

058

1

ASU 058 All Russian MO 1965 line passes through incenter

A circle is circumscribed around the triangle

ABC

. Chords, from the midpoint of the arc

AC

to the midpoints of the arcs

AB

and

BC

, intersect sides

[AB]

and

[BC]

in the points

D

and

E

. Prove that

(DE)

is parallel to

(AC)

and passes through the centre of the inscribed circle.

057

1

ASU 057 All Russian MO 1965 game strategy, board 3x3 and 9 cards

Given a board

3\times3

and

9

cards with some numbers (known to the players). Two players, in turn, put those cards on the board. The first wins if the sum of the numbers in the first and the third row is greater than in the first and the third column. Prove that it doesn't matter what numbers are on the cards, but if the first plays the best way, the second can not win.

056

1

ASU 056 All Russian MO 1965 min sum of products of x_1,x_2,...,x_n

a) Each of the numbers

x_1,x_2,...,x_n

can be

1, 0

, or

-1

. What is the minimal possible value of the sum of all products of couples of those numbers.
b) Each absolute value of the numbers

x_1,x_2,...,x_n

doesn't exceed

1

. What is the minimal possible value of the sum of all products of couples of those numbers.

1965 All Russian Mathematical Olympiad

Subcontests

ASU 071 All Russian MO 1965 spaceship finds the inhabitant on planet

ASU 070 All Russian MO 1965 sum of lengths of polyhedron edges more than

ASU 069 All Russian MO 1965 rocker hits the airplane-spy

ASU 068 All Russian MO 1965 n = px + qy, relative primes p,q

ASU 067 All Russian MO 1965 members of subcommittees

ASU 066 All Russian MO 1965 tourist in Moscow

ASU 065 All Russian MO 1965 quasi-rounding n numbers

ASU 064 All Russian MO 1965, 1965 points inside the unit square

ASU 063 All Russian MO 1965 x_{i,j}+x_{j,k}+x_{k,i}=0

ASU 062 All Russian MO 1965 max length of segment, incircle related

ASU 061 All Russian MO 1965 100 men help the police, 3 every evening

ASU 060 All Russian MO 1965 lighthouse enlights a ship

ASU 059 All Russian MO 1965 first 3 digits = last 3 digits, divisible by 13

ASU 058 All Russian MO 1965 line passes through incenter

ASU 057 All Russian MO 1965 game strategy, board 3x3 and 9 cards

ASU 056 All Russian MO 1965 min sum of products of x_1,x_2,...,x_n