MathDB

1

Arithmetic progression [Iran TST 2010]

f:\mathbb N\rightarrow\mathbb N

be a non-decreasing function and let

n

be an arbitrary natural number. Suppose that there are prime numbers

p_1,p_2,\dots,p_n

and natural numbers

s_1,s_2,\dots,s_n

such that for each

1\leq i\leq n

the set

\{f(p_ir+s_i)|r=1,2,\dots\}

is an infinite arithmetic progression. Prove that there is a natural number

a

such that

f(a+1), f(a+2), \dots, f(a+n)

form an arithmetic progression.

2

1

Non-decreasing function [Iran TST 2010]

Find all non-decreasing functions

f:\mathbb R^+\cup\{0\}\rightarrow\mathbb R^+\cup\{0\}

such that for each

x,y\in \mathbb R^+\cup\{0\}

f\left(\frac{x+f(x)}2+y\right)=2x-f(x)+f(f(y)).

3

1

Two-variable polynomial [Iran TST 2010]

Find all two-variable polynomials

p(x,y)

such that for each

a,b,c\in\mathbb R

:

p(ab,c^2+1)+p(bc,a^2+1)+p(ca,b^2+1)=0

4

1

Distance preserving map between trees [Iran TST 2010]

S,T

are two trees without vertices of degree 2. To each edge is associated a positive number which is called length of this edge. Distance between two arbitrary vertices

v,w

in this graph is defined by sum of length of all edges in the path between

v

and

w

. Let

f

be a bijective function from leaves of

S

to leaves of

T

, such that for each two leaves

u,v

of

S

, distance of

u,v

in

S

is equal to distance of

f(u), f(v)

in

T

. Prove that there is a bijective function

g

from vertices of

S

to vertices of

T

such that for each two vertices

u,v

of

S

, distance of

u,v

in

S

is equal to distance of

g(u)

and

g(v)

in

T

.

5

1

Equality of angles [Iran TST 2010]

Circles

W_1,W_2

intersect at

P,K

.

XY

is common tangent of two circles which is nearer to

P

and

X

is on

W_1

and

Y

is on

W_2

.

XP

intersects

W_2

for the second time in

C

and

YP

intersects

W_1

in

B

. Let

A

be intersection point of

BX

and

CY

. Prove that if

Q

is the second intersection point of circumcircles of

ABC

and

AXY

\angle QXA=\angle QKP

6

1

APT and KPC are tangent [Iran TST 2010]

Let

M

be an arbitrary point on side

BC

of triangle

ABC

.

W

is a circle which is tangent to

AB

and

BM

at

T

and

K

and is tangent to circumcircle of

AMC

at

P

. Prove that if

TK||AM

, circumcircles of

APT

and

KPC

are tangent together.

7

1

polygon cuts itself [Iran TST 2010]

Without lifting pen from paper, we draw a polygon in such away that from every two adjacent sides one of them is vertical. In addition, while drawing the polygon all vertical sides have been drawn from up to down. Prove that this polygon has cut itself.

8

1

Isoceles triangle [Iran TST 2010]

Let

ABC

an isosceles triangle and

BC>AB=AC

.

D,M

are respectively midpoints of

BC, AB

.

X

is a point such that

BX\perp AC

and

XD||AB

.

BX

and

AD

meet at

H

. If

P

is intersection point of

DX

and circumcircle of

AHX

(other than

X

), prove that tangent from

A

to circumcircle of triangle

AMP

is parallel to

BC

.

9

1

Concave Sequence [Iran TST 2010]

Sequence of real numbers

a_0,a_1,\dots,a_{1389}

are called concave if for each

0<i<1389

,

a_i\geq\frac{a_{i-1}+a_{i+1}}2

. Find the largest

c

such that for every concave sequence of non-negative real numbers:

\sum_{i=0}^{1389}ia_i^2\geq c\sum_{i=0}^{1389}a_i^2

10

1

Path from sides [Iran TST 2010]

In every

1\times1

square of an

m\times n

table we have drawn one of two diagonals. Prove that there is a path including these diagonals either from left side to the right side, or from the upper side to the lower side.

11

1

Cyclic Quadrilateral [Iran TST 2010]

Let

O, H

be circumcenter and orthogonal center of triangle

ABC

.

M,N

are midpoints of

BH

and

CH

.

BB'

is diagonal of circumcircle. If

HONM

is a cyclic quadrilateral, prove that

B'N=\frac12AC

.

12

1

Sum of digits in [Iran TST 2010]

Prove that for each natural number

m

, there is a natural number

N

such that for each

b

that

2\leq b\leq1389

sum of digits of

N

in base

b

is larger than

m

.

2010 Iran Team Selection Test

Subcontests

Arithmetic progression [Iran TST 2010]

Non-decreasing function [Iran TST 2010]

Two-variable polynomial [Iran TST 2010]

Distance preserving map between trees [Iran TST 2010]

Equality of angles [Iran TST 2010]

APT and KPC are tangent [Iran TST 2010]

polygon cuts itself [Iran TST 2010]

Isoceles triangle [Iran TST 2010]

Concave Sequence [Iran TST 2010]

Path from sides [Iran TST 2010]

Cyclic Quadrilateral [Iran TST 2010]

Sum of digits in [Iran TST 2010]