Challenging Recreational Mathematics: ▪ Iran Team Selection Test 2013

Τετάρτη 8 Μαΐου 2013

▪ Iran Team Selection Test 2013

Ημέρα 1η

1 In acute-angled triangle

, let

be the foot of perpendicular from

and also suppose that

and

are excenters oposite to the side

in triangles

and

. If

is the point that incircle touches

, prove that

are concyclic.

2 Find the maximum number of subsets from

such that for any two of them like

then

. (Here

is the number of elements of the set

3 For nonnegative integers

and

, define the sequence

of real numbers as follows. Set

and for every natural number

, set

and

. Then for

, define

Prove that for every natural number

, all the roots of the polynomial

are real.

Ημέρα 2η

and

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

cells in one of the

directions, then it turns

degrees clockwise and after that moves

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

5 Do there exist natural numbers

and

such that

is divisible by

Proposed by Mahan Malihi

6 Points

and

lie on line

in this order. Two circular arcs

and

, which both lie on one side of line

, pass through points

and

and two circular arcs

and

pass through points

and

such that

is tangent to

and

is tangent to

. Prove that the common external tangent of

and

and the common external tangent of

and

meet each other on line

Proposed by Ali Khezeli

TST 2

Ημέρα 1η

7 Nonnegative real numbers

and

are such that

Between all the matrices with nonnegative entries which sum of their

-th row's entries is

and sum of their

-th column's entries is

, Find the maximum of sum of the entries on the main diagonal.

8 Find all Arithmetic progressions

of natural numbers for which there exists natural number

such that for every

9 find all functions

such that

is increasing and also:

Ημέρα 2η

10 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)

11 Let