MathDB

1

The perpendicular bisector of the bisector

ABC

,

AD

and

AH

are the angle bisector and the altitude of vertex

A

, respectively. The perpendicular bisector of

AD

, intersects the semicircles with diameters

AB

and

AC

which are drawn outside triangle

ABC

in

X

and

Y

, respectively. Prove that the quadrilateral

XYDH

is concyclic.
Proposed by Mahan Malihi

16

1

f(m)+f(n)+f(f(m^2+n^2))=1

The function

f:\mathbb Z \to \mathbb Z

has the property that for all integers

m

and

n

f(m)+f(n)+f(f(m^2+n^2))=1.

We know that integers

a

and

b

exist such that

f(a)-f(b)=3

. Prove that integers

c

and

d

can be found such that

f(c)-f(d)=1

.
Proposed by Amirhossein Gorzi

18

1

Tiling a room

A special kind of parallelogram tile is made up by attaching the legs of two right isosceles triangles of side length

1

. We want to put a number of these tiles on the floor of an

n\times n

room such that the distance from each vertex of each tile to the sides of the room is an integer and also no two tiles overlap. Prove that at least an area

n

of the room will not be covered by the tiles.
Proposed by Ali Khezeli

15

1

sequence with divisors

a) Does there exist a sequence

a_1<a_2<\dots

of positive integers, such that there is a positive integer

N

that

\forall m>N

,

a_m

has exactly

d(m)-1

divisors among

a_i

s?
b) Does there exist a sequence

a_1<a_2<\dots

of positive integers, such that there is a positive integer

N

that

\forall m>N

,

a_m

has exactly

d(m)+1

divisors among

a_i

s?

14

1

4n Right Angles

we are given

n

rectangles in the plane. Prove that between

4n

right angles formed by these rectangles there are at least

[4\sqrt n]

distinct right angles.

13

1

Centroid Inside Triangle

P

is an arbitrary point inside acute triangle

ABC

. Let

A_1,B_1,C_1

be the reflections of point

P

with respect to sides

BC,CA,AB

. Prove that the centroid of triangle

A_1B_1C_1

lies inside triangle

ABC

.

12

1

Prove that some three points lie on a line

Let

ABCD

be a cyclic quadrilateral that inscribed in the circle

\omega

.Let

I_{1},I_{2}

and

r_{1},r_{2}

be incenters and radii of incircles of triangles

ACD

and

ABC

,respectively.assume that

r_{1}=r_{2}

. let

\omega'

be a circle that touches

AB,AD

and touches

\omega

at

T

. tangents from

A,T

to

\omega

meet at the point

K

.prove that

I_{1},I_{2},K

lie on a line.

11

1

Inequality about sides of a triangle

Let

a,b,c

be sides of a triangle such that

a\geq b \geq c

. prove that:

\sqrt{a(a+b-\sqrt{ab})}+\sqrt{b(a+c-\sqrt{ac})}+\sqrt{c(b+c-\sqrt{bc})}\geq a+b+c

10

1

Weighted graph

On each edge of a graph is written a real number,such that for every even tour of this graph,sum the edges with signs alternatively positive and negative is zero.prove that one can assign to each of the vertices of the graph a real number such that sum of the numbers on two adjacent vertices is the number on the edge between them.(tour is a closed path from the edges of the graph that may have repeated edges or vertices)

9

1

functional equation

find all functions

f,g:\mathbb{R}^{+} \rightarrow \mathbb{R}^{+}

such that

f

is increasing and also:

f(f(x)+2g(x)+3f(y))=g(x)+2f(x)+3g(y)

g(f(x)+y+g(y))=2x-g(x)+f(y)+y

8

1

Find all Arithmetic progressions

a_{1},a_{2},...

of natural numbers for which there exists natural number

N>1

such that for every

k\in \mathbb{N}

:

a_{1}a_{2}...a_{k}\mid a_{N+1}a_{N+2}...a_{N+k}

7

1

Matrices with defined property

Nonnegative real numbers

p_{1},\ldots,p_{n}

and

q_{1},\ldots,q_{n}

are such that

p_{1}+\cdots+p_{n}=q_{1}+\cdots+q_{n}

Among all the matrices with nonnegative entries having

p_i

as sum of the

i

-th row's entries and

q_j

as sum of the

j

-th column's entries, find the maximum sum of the entries on the main diagonal.

6

1

Concurrency of two common external tangents

Points

A, B, C

and

D

lie on line

l

in this order. Two circular arcs

C_1

and

C_2

, which both lie on one side of line

l

, pass through points

A

and

B

and two circular arcs

C_3

and

C_4

pass through points

C

and

D

such that

C_1

is tangent to

C_3

and

C_2

is tangent to

C_4

. Prove that the common external tangent of

C_2

and

C_3

and the common external tangent of

C_1

and

C_4

meet each other on line

l

.
Proposed by Ali Khezeli

4

1

Philosopher's Chess

m

and

n

are two nonnegative integers. In the Philosopher's Chess, The chessboard is an infinite grid of identical regular hexagons and a new piece named the Donkey moves on it as follows:
Starting from one of the hexagons, the Donkey moves

m

cells in one of the

6

directions, then it turns

60

degrees clockwise and after that moves

n

cells in this new direction until it reaches it's final cell.
At most how many cells are in the Philosopher's chessboard such that one cannot go from anyone of them to the other with a finite number of movements of the Donkey?
Proposed by Shayan Dashmiz

3

1

Roots of a polynomial defined by a sequence are real

For nonnegative integers

m

and

n

, define the sequence

a(m,n)

of real numbers as follows. Set

a(0,0)=2

and for every natural number

n

, set

a(0,n)=1

and

a(n,0)=2

. Then for

m,n\geq1

, define

a(m,n)=a(m-1,n)+a(m,n-1).

Prove that for every natural number

k

, all the roots of the polynomial

P_{k}(x)=\sum_{i=0}^{k}a(i,2k+1-2i)x^{i}

are real.

2

1

|B-A| >= 3 for any subsets A,B

Find the maximum number of subsets from

\left \{ 1,...,n \right \}

such that for any two of them like

A,B

if

A\subset B

then

\left | B-A \right |\geq 3

. (Here

\left | X \right |

is the number of elements of the set

X

.)

1

Prove that I,J,P,H are concyclic

In acute-angled triangle

ABC

, let

H

be the foot of perpendicular from

A

to

BC

and also suppose that

J

and

I

are excenters oposite to the side

AH

in triangles

ABH

and

ACH

. If

P

is the point that incircle touches

BC

, prove that

I,J,P,H

are concyclic.

2013 Iran Team Selection Test

Subcontests

The perpendicular bisector of the bisector

f(m)+f(n)+f(f(m^2+n^2))=1

Tiling a room

sequence with divisors

4n Right Angles

Centroid Inside Triangle

Prove that some three points lie on a line

Inequality about sides of a triangle

Weighted graph

functional equation

Find all Arithmetic progressions

Matrices with defined property

Concurrency of two common external tangents

Philosopher's Chess

Roots of a polynomial defined by a sequence are real

|B-A| >= 3 for any subsets A,B

Prove that I,J,P,H are concyclic

2013 Iran Team Selection Test

Subcontests

The perpendicular bisector of the bisector

f(m)+f(n)+f(f(m^2+n^2))=1

Tiling a room

sequence with divisors

4n Right Angles

Centroid Inside Triangle

Prove that some three points lie on a line

Inequality about sides of a triangle

Weighted graph

functional equation

Find all Arithmetic progressions

Matrices with defined property

Concurrency of two common external tangents

Philosopher's Chess

Roots of a polynomial defined by a sequence are real

|B-A| &gt;= 3 for any subsets A,B

Prove that I,J,P,H are concyclic

|B-A| >= 3 for any subsets A,B