2002 Iran MO (3rd Round)

Part of Iran MO (3rd Round)

Subcontests

(25)

1

Ant

An ant walks on the interior surface of a cube, he moves on a straight line. If ant reaches to an edge the he moves on a straight line on cube's net. Also if he reaches to a vertex he will return his path. a) Prove that for each beginning point ant can has infinitely many choices for his direction that its path becomes periodic. b) Prove that if if the ant starts from point

A

and its path is periodic, then for each point

B

if ant starts with this direction, then his path becomes periodic.

1

Old Persia

15000 years ago Tilif ministry in Persia decided to define a code for

n\geq2

cities. Each code is a sequence of

0,1

such that no code start with another code. We know that from

2^{m}

calls from foreign countries to Persia

2^{m-a_{i}}

of them where from the

i

\sum_{i=1}^{n}\frac1{2^{a_{i}}}=1

l_{i}

be length of code assigned to

i

-th city. Prove that

\sum_{i=1}^{n}\frac{l_{i}}{2^{i}}

\forall i,\ l_{i}=a_{i}

1

radii are equal

Excircle of triangle

ABC

corresponding vertex

A

, is tangent to

BC

P

AP

intersects circumcircle of

ABC

D

r(PCD)=r(PBD)

r(PCD)

r(PBD)

are inradii of triangles

PCD

PBD

1

Collinear

I

is incenter of triangle

ABC

ABC

AB,AC

X,Y

XI

intersects incircle at

M

CM\cap AB=X'

L

is a point on the segment

X'C

X'L=CM

A,L,I

are collinear iff

AB=AC

1

Two lines

Find the smallest natural number

n

that the following statement holds : Let

A

be a finite subset of

\mathbb R^{2}

n

A

there are two lines including these

n

points. All of the points lie on two lines.

1

Eventually linear

S

\mathbb N

is eventually linear iff there are

k,N\in\mathbb N

n>N,n\in S\Longleftrightarrow k|n

S

\mathbb N

that is closed under addition. Prove that

S

is eventually linear.

1

Permutation

f,g

are two permutations of set

X=\{1,\dots,n\}

f,g

have common points iff there is a

k\in X

f(k)=g(k)

m>\frac{n}{2}

, prove that there are

m

f_{1},f_{2},\dots,f_{m}

X

that for each permutation

f\in X

, there is an index

i

f,f_{i}

have common points. b) Prove that if

m\leq\frac{n}{2}

, we can not find permutations

f_{1},f_{2},\dots,f_{m}

satisfying the above condition.

1

Graph

We have a bipartite graph

G

X

Y

). We orient each edge arbitrarily. Hessam chooses a vertex at each turn and reverse the orientation of all edges that

v

is one of their endpoint. Prove that with these steps we can reach to a graph that for each vertex

v

X

\deg^{+}(v)\geq \deg^{-}(v)

and for each vertex in part

Y

\deg^{+}v\leq \deg^{-}v

1

Policeman and thief

m\times n

table there is a policeman in cell

(1,1)

, and there is a thief in cell

(i,j)

. A move is going from a cell to a neighbor (each cell has at most four neighbors). Thief makes the first move, then the policeman moves and ... For which

(i,j)

the policeman can catch the thief?

1

Nagelian point

H,I,O,N

are orthogonal center, incenter, circumcenter, and Nagelian point of triangle

ABC

I_{a},I_{b},I_{c}

are excenters of

ABC

corresponding vertices

A,B,C

S

O

HS

. Prove that centroid of triangles

I_{a}I_{b}I_{c}

SIN

1

Triangle problem

\omega

is circumcirlce of triangle

ABC

. We draw a line parallel to

BC

that intersects

AB,AC

E,F

\omega

U,V

M

BC

\omega'

be circumcircle of

UMV

R(ABC)=R(UMV)

ME

\omega'

T

FT

\omega'

S

EF

is tangent to circumcircle of

MCS

1

a_n is integer

a_{n}

is a sequence that

a_{1}=1,a_{2}=2,a_{3}=3

a_{n+1}=a_{n}-a_{n-1}+\frac{a_{n}^{2}}{a_{n-2}}

Prove that for each natural

n

a_{n}

1

Function

f: \mathbb R\longrightarrow\mathbb R^{+}

is a non-decreasing function. Prove that there is a point

a\in\mathbb R

f(a+\frac1{f(a)})<2f(a)

1

a,b,c

a,b,c\in\mathbb R^{n}, a+b+c=0

\lambda>0

\prod_{cycle}\frac{|a|+|b|+(2\lambda+1)|c|}{|a|+|b|+|c|}\geq(2\lambda+3)^{3}

1

Tricky triangle problem

M

N

be points on the side

BC

ABC

, with the point

M

lying on the segment

BN

, such that BM \equal{} CN. Let

P

Q

be points on the segments

AN

AM

, respectively, such that \measuredangle PMC \equal{}\measuredangle MAB and \measuredangle QNB \equal{}\measuredangle NAC. Prove that \measuredangle QBC \equal{}\measuredangle PCB.

1

Easy

M

BC

P

is an arbitary point on

BC

C_{1}

is tangent to big circle.Suppose radius of

C_{1}

r_{1}

C_{4}

is equal to radius of

C_{1}

C_{4}

BC

C_{2}

C_{3}

are tangent to big circle and line

BC

C_{4}

. http://aycu01.webshots.com/image/4120/2005120338156776027_rs.jpg Prove :

r_{1}+r_{2}+r_{3}=R

R

radius of big circle)

1

Easy one

Find all polynomials

p

with real coefficients that if for a real

a

p(a)

is integer then

a

1

Nice one

Let A be be a point outside the circle C, and AB and AC be the two tangents from A to this circle C. Let L be an arbitrary tangent to C that cuts AB and AC in P and Q. A line through P parallel to AC cuts BC in R. Prove that while L varies, QR passes through a fixed point. :)

1

Prove they are on aline [about mixtilinear incircles]

A,B,C

\mathcal C

I

ABC

D

is midpoint of arc

BAC

W

is a circle that is tangent to

AB

AC

\mathcal C

P

W

\mathcal C

P

I

D

1

They are concurrent

C_{1}

C_{2}

are tangent to each other at

K

and are tangent to circle

C

M

N

. External tangent of

C_{1}

C_{2}

C

A

B

AK

BK

intersect with circle

C

E

F

respectively. If AB is diameter of

C

EF

MN

OK

are concurrent. (

O

is center of circle

C

1

What are the angles

ABC

AD

is angle bisector (

D

BC

AB+AD=CD

AC+AD=BC

, what are the angles of

ABC

1

Sequence

a_{n}

n

is integer) is a sequence from positive reals that

a_{n}\geq \frac{a_{n+2}+a_{n+1}+a_{n-1}+a_{n-2}}4

a_{n}

1

Another functional equation

Find all continious

f: \mathbb R\longrightarrow\mathbb R

x,y

f(x)+f(y)+f(xy)=f(x+y+xy)

1

Sequence

a_{0}=2,a_{1}=1

n\geq 1

a_{n+1}=a_{n}+a_{n-1}

m

is an even number and

p

is prime number such that

p

a_{m}-2

p

a_{m+1}-1

1

Iran 2002

a,b,c

a^{2}+b^{2}+c^{2}+abc=4

a+b+c \leq3