MathDB

1

concyclic , touchpoints of incircle related

O

is tangent to the sides

AB

,

BC

,

AC

of a triangle

ABC

at points

E,F,D

respectively. The lines

AO

and

CO

meet

EF

at points

N

and

M

. Prove that the circumcircle of triangle

OMN

and points

O

and

D

lie on a line.

11.6

1

(x-1)f )( x+1)/(x−1) )- f(x) = x All-Russian MO 1994 Regional (R4) 11.6

Find all functions satisfying the equality

(x-1)f \left(\dfrac{x+1}{x-1}\right)- f(x) = x

for all

x \ne 1

.

11.2

1

at most 1 phone call All-Russian MO 1994 Regional (R4) 11.2

It was noted that during one day in a town, each person made at most one phone call. Prove that the people in the town can be divided into three groups such that no two persons in the same group talked by phone that day.

10.6

1

P(19) = P(94) = 1994. All-Russian MO 1994 Regional (R4) 10.6

Find the free coefficient of the polynomial

P(x)

with integer coefficients, knowing that it is less than

1000

in absolute value and that

P(19) = P(94) = 1994

.

10.1

1

7 equal pails with water All-Russian MO 1994 Regional (R4) 10.1

We have seven equal pails with water, filled to one half, one third, one quarter, one fifth, one eighth, one ninth, and one tenth, respectively. We are allowed to pour water from one pail into another until the first pail empties or the second one fills to the brim. Can we obtain a pail that is filled to
(a) one twelfth, (b) one sixth
after several such steps?

9.7

1

p^2+qs, p^2 +qr perfect squares - All-Russian MO 1994 Regional (R4) 9.7

Find all prime numbers

p,q,r,s

such that their sum is a prime number and

p^2+qs

and

p^2 +qr

are squares of integers.

9.5

1

cx^5 +bx+a = 0, ax^5 + bx^4 + c = 0 All-Russian MO 1994 Regional (R4) 9.5

The equation

ax^5 + bx^4 + c = 0

has three distinct roots. Show that so does the equation

cx^5 +bx+a = 0

.

9.4

1

liars and truth-lovers All-Russian MO 1994 Regional (R4) 9.4

On the world conference of parties of liars and truth-lovers there were

32

participants which were sitting in four rows with

8

chairs each. During a break each participant claimed that among his neighbors (by row or column) there are members of both parties. It is known that liars always lie, whereas truth-lovers always tell truth. What is the smallest number of liars at the conference for which this situation is possible?

9.3

1

p(n)= only digits 1 - All-Russian MO 1994 Regional (R4) 9.3

Does there exist a quadratic trinomial

p(x)

with integer coefficients such that, for every natural number

n

whose decimal representation consists of digits

1

,

p(n)

also consists only of digits

1

?

9.2

1

4 cities - All-Russian MO 1994 Regional (R4) 9.2

Cities

A,B,C,D

are positioned in such a way that

A

is closer to

C

than to

D

, and

B

is closer to

C

than to

D

. Prove that every point on the straight road from

A

to

B

is closer to

C

than to

D

.

9.1

1

milk and honey - All-Russian MO 1994 Regional (R4) 9.1

One day, Rabbit was about to go for a meeting with Donkey, but Winnie the Pooh and Duck unexpectedly came to his home. Being well-bred, Rabbit offered the guests some refreshments. Pooh tied Duck’s mouth by a napkin and ate

10

pots of honey and

22

cups of condensed milk alone, whereby he needed two minutes for each pot of honey and

1

minute for each cup of milk. Knowing that there was nothing sweet left in the house, Pooh released the Duck. Afflicted Rabbit observed that he wouldn’t have been late for the meeting with Donkey if Pooh had shared the refreshments with Duck. Knowing that Duck needs

5

minutes for a pot of honey and

3

minutes for a cup of milk, he computed the time the guests would have needed to devastate his supplies. What is that time?

11.7

1

SO _|_ (A_1B_1C), regular triangular pyramid SABC , sphere ( A,B,C_1,S)

Points

A_1

,

B_1

and

C_1

are taken on the respective edges

SA

,

SB

,

SC

of a regular triangular pyramid

SABC

so that the planes

A_1B_1C_1

and

ABC

are parallel. Let

O

be the center of the sphere passing through

A

,

B

,

C_1

and

S

. Prove that the line

SO

is perpendicular to the plane

A_1B_1C

.

11.1

1

Russian trig inequality

Prove that for all

x \in \left( 0, \frac{\pi}{3} \right)

inequality

sin2x+cosx>1

holds.

10.3

1

circle

A circle with center O is inscribed in a quadrilateral ABCD and touches its non-parallel sides BC and AD at E and F respectively. The lines AO and DO meet the segment EF at K and N respectively, and the lines BK and CN meet at M. Prove that the points O,K,M and N lie on a circle.

11.8

1

bug traveling in the circle

Points

A_1,A_2, ... ,A_n

inside a circle and points

B_1,B_2,...,B_n

on its boundary are positioned so that the segments

A_1B_1,A_2B_2, ... ,A_nB_n

do not intersect. A bug can go from point

A_i

to

A_j

if the segment

A_iA_j

does not intersect any segment

A_kB_k

,

k \neq i, j

. Prove that the bug can go from any point

A_p

to any point

A_q

in a finite number of steps.

10.8

1

naming streets

In the Flower-city there are

n

squares and

m

streets, where m \geq n \plus{} 1. Each street connects two squares and does not pass through other squares. According to a tradition in the city, each street is named either blue or red. Every year, a square is selected and the names of all streets emanating from that square are changed. Show that the streets can be initially named in such a way that, no matter how the names will be changed, the streets will never all have the same name.

10.4

1

placing trominoes

A rectangle of size

m \times n

has been filled completely by trominoes (a tromino is an L-shape consisting of 3 unit squares). There are four ways to place a tromino 1st way: let the "corner" of the L be on top left 2nd way: let the "corner" of the L be on top right 3rd way: let the "corner" of the L be on bottom left 4th way: let the "corner" of the L be on bottom right Prove that the difference between the number of trominoes placed in the 1st and the 4th way is divisible by

3

.

9.8

1

dividing pupils

There are

16

pupils in a class. Every month, the teacher divides the pupils into two groups. Find the smallest number of months after which it will be possible that every two pupils were in two different groups during at least one month.

10.2

1

distinct roots in quadratic and quartic

The equation x^2 \plus{} ax \plus{} b \equal{} 0 has two distinct real roots. Prove that the equation x^4 \plus{} ax^3 \plus{} (b \minus{} 2)x^2 \minus{} ax \plus{} 1 \equal{} 0 has four distinct real roots.

11.4

1

moving stones

On the vertices of a convex

n

-gon are put

m

stones,

m > n

. In each move we can choose two stones standing at the same vertex and move them to the two distinct adjacent vertices. After

N

moves the number of stones at each vertex was the same as at the beginning. Prove that

N

is divisible by

n

.

10.7

1

inequality in pentagon

In a convex pentagon

ABCDE

side

AB

is perpendicular to

CD

and side

BC

is perpendicular to

DE

. Prove that if AB \equal{} AE \equal{} ED \equal{} 1, then BC \plus{} CD < 1.

9.6

1

two tangent circles

Point

P

is taken inside a right angle

KLM

. A circle

S_1

with center

O_1

is tangent to the rays

LK,LP

of angle

KLP

at

A,D

respectively. A circle

S_2

with center

O_2

is tangent to the rays of angle

MLP

, touching

LP

at

B

. Suppose

A,B,O_1

are collinear. Let

O_2D,KL

meet at

C

. Prove that

BC

bisects angle

ABD

.

1994 All-Russian Olympiad Regional Round

Subcontests

concyclic , touchpoints of incircle related

(x-1)f )( x+1)/(x−1) )- f(x) = x All-Russian MO 1994 Regional (R4) 11.6

at most 1 phone call All-Russian MO 1994 Regional (R4) 11.2

P(19) = P(94) = 1994. All-Russian MO 1994 Regional (R4) 10.6

7 equal pails with water All-Russian MO 1994 Regional (R4) 10.1

p^2+qs, p^2 +qr perfect squares - All-Russian MO 1994 Regional (R4) 9.7

cx^5 +bx+a = 0, ax^5 + bx^4 + c = 0 All-Russian MO 1994 Regional (R4) 9.5

liars and truth-lovers All-Russian MO 1994 Regional (R4) 9.4

p(n)= only digits 1 - All-Russian MO 1994 Regional (R4) 9.3

4 cities - All-Russian MO 1994 Regional (R4) 9.2

milk and honey - All-Russian MO 1994 Regional (R4) 9.1

SO _|_ (A_1B_1C), regular triangular pyramid SABC , sphere ( A,B,C_1,S)

Russian trig inequality

circle

bug traveling in the circle

naming streets

placing trominoes

dividing pupils

distinct roots in quadratic and quartic

moving stones

inequality in pentagon

two tangent circles

1994 All-Russian Olympiad Regional Round

Subcontests

concyclic , touchpoints of incircle related

(x-1)f )( x+1)/(x&minus;1) )- f(x) = x All-Russian MO 1994 Regional (R4) 11.6

at most 1 phone call All-Russian MO 1994 Regional (R4) 11.2

P(19) = P(94) = 1994. All-Russian MO 1994 Regional (R4) 10.6

7 equal pails with water All-Russian MO 1994 Regional (R4) 10.1

p^2+qs, p^2 +qr perfect squares - All-Russian MO 1994 Regional (R4) 9.7

cx^5 +bx+a = 0, ax^5 + bx^4 + c = 0 All-Russian MO 1994 Regional (R4) 9.5

liars and truth-lovers All-Russian MO 1994 Regional (R4) 9.4

p(n)= only digits 1 - All-Russian MO 1994 Regional (R4) 9.3

4 cities - All-Russian MO 1994 Regional (R4) 9.2

milk and honey - All-Russian MO 1994 Regional (R4) 9.1

SO _|_ (A_1B_1C), regular triangular pyramid SABC , sphere ( A,B,C_1,S)

Russian trig inequality

circle

bug traveling in the circle

naming streets

placing trominoes

dividing pupils

distinct roots in quadratic and quartic

moving stones

inequality in pentagon

two tangent circles

(x-1)f )( x+1)/(x−1) )- f(x) = x All-Russian MO 1994 Regional (R4) 11.6