2006 Iran Team Selection Test

Part of Iran Team Selection Test

Subcontests

(6)

2

Question 6

G

be a tournoment such that it's edges are colored either red or blue. Prove that there exists a vertex of

G

v

with the property that, for every other vertex

u

there is a mono-color directed path from

v

u

Question 6

Suppose we have a simple polygon (that is it does not intersect itself, but not necessarily convex). Show that this polygon has a diameter which is completely inside the polygon and the two arcs it creates on the polygon perimeter (the two arcs have 2 vertices in common) both have at least one third of the vertices of the polygon.

2

Question 5

ABC

be a triangle such that it's circumcircle radius is equal to the radius of outer inscribed circle with respect to

A

. Suppose that the outer inscribed circle with respect to

A

BC,AC,AB

M,N,L

O

(Center of circumcircle) is the orthocenter of

MNL

Question 5

ABC

be an acute angle triangle. Suppose that

D,E,F

are the feet of perpendicluar lines from

A,B,C

BC,CA,AB

P,Q,R

be the feet of perpendicular lines from

A,B,C

EF,FD,DE

2(PQ+QR+RP)\geq DE+EF+FD

2

Question 4

x_1,x_2,\ldots,x_n

be real numbers. Prove that

\sum_{i,j=1}^n |x_i+x_j|\geq n\sum_{i=1}^n |x_i|

Question: a_1+a_2+...+a_n | a_1^i+a_2^i+...+a_n^i

n

be a fixed natural number. Find all

n

tuples of natural pairwise distinct and coprime numbers like

a_1,a_2,\ldots,a_n

1\leq i\leq n

a_1+a_2+\ldots+a_n|a_1^i+a_2^i+\ldots+a_n^i

2

Question 1

p

is a prime number. Find all natural numbers

n

p|\varphi(n)

a

(a,n)=1

n|a^{\frac{\varphi(n)}{p}}-1

Question 1

n

points in the plane, no three on a line. We call

k

of them good if they form a convex polygon and there is no other point in the convex polygon. Suppose that for a fixed

k

k

c_k

. Show that the following sum is independent of the structure of points and only depends on

n

\sum_{i=3}^n (-1)^i c_i

2

Question 3

ABC

is a triangle with

M

the midpoint of

BC

AM

intersects the incircle at

K,L

. We draw parallel line from

K

L

BC

and name their second intersection point with incircle

X

Y

AX

AY

BC

P

Q

BP=CQ

Question 3

l,m

be two parallel lines in the plane. Let

P

be a fixed point between them. Let

E,F

be variable points on

l,m

such that the angle

EPF

is fixed to a number like

\alpha

0<\alpha<\frac{\pi}2

EPF

we mean the directed angle) Show that there is another point (not

P

) such that it sees the segment

EF

with a fixed angle too.

2

Question 2

n

coins are available that their mass is unknown. We have a pair of balances and every time we can choose an even number of coins and put half of them on one side of the balance and put another half on the other side, therefore a comparison will be done. Our aim is determining that the mass of all coins is equal or not. Show that at least

n-1

comparisons are required.

Question 2

n

be a fixed natural number. a) Find all solutions to the following equation :

\sum_{k=1}^n [\frac x{2^k}]=x-1

b) Find the number of solutions to the following equation (

m

is a fixed natural) :

\sum_{k=1}^n [\frac x{2^k}]=x-m