MathDB

1

Polynomials commuting with z^2+c

c\in\mathbb C

and A_c \equal{} \{p\in \mathbb C[z]|p(z^2 \plus{} c) \equal{} p(z)^2 \plus{} c\}. a) Prove that for each

c\in C

,

A_c

is infinite. b) Prove that if

p\in A_1

, and p(z_0) \equal{} 0, then

|z_0| < 1.7

. c) Prove that each element of

A_c

is odd or even. Let f_c \equal{} z^2 \plus{} c\in \mathbb C[z]. We see easily that B_c: \equal{} \{z,f_c(z),f_c(f_c(z)),\dots\} is a subset of

A_c

. Prove that in the following cases A_c \equal{} B_c. d)

|c| > 2

. e)

c\in \mathbb Q\backslash\mathbb Z

. f)

c

is a non-algebraic number g)

c

is a real number and c\not\in [ \minus{} 2,\frac14].

28

1

Points in space with non-acute angles between them

There are

n

points in

\mathbb R^3

such that every three form an acute angled triangle. Find maximum of

n

.

27

1

Square sets

S\subset\mathbb N

is called a square set, iff for each

x,y\in S

, xy\plus{}1 is square of an integer. a) Is

S

finite? b) Find maximum number of elements of

S

.

26

1

Median has fixed length

Circles

C_1,C_2

intersect at

P

. A line

\Delta

is drawn arbitrarily from

P

and intersects with

C_1,C_2

at

B,C

. What is locus of

A

such that the median of

AM

of triangle

ABC

has fixed length

k

.

25

1

DD', EE' and FF' intersect at Q

Let

A,B,C,Q

be fixed points on plane.

M,N,P

are intersection points of

AQ,BQ,CQ

with

BC,CA,AB

.

D',E',F'

are tangency points of incircle of

ABC

with

BC,CA,AB

. Tangents drawn from

M,N,P

(not triangle sides) to incircle of

ABC

make triangle

DEF

. Prove that

DD',EE',FF'

intersect at

Q

.

24

1

a fixed point on circle AMN

A,B

are fixed points. Variable line

l

passes through the fixed point

C

. There are two circles passing through

A,B

and tangent to

l

at

M,N

. Prove that circumcircle of

AMN

passes through a fixed point.

23

1

Homogeneous linear recursive sequences

Find all homogeneous linear recursive sequences such that there is a

T

such that a_n\equal{}a_{n\plus{}T} for each

n

.

22

1

When does this sequence has integer values?

Let a_1\equal{}a_2\equal{}1 and a_{n\plus{}2}\equal{}\frac{n(n\plus{}1)a_{n\plus{}1}\plus{}n^2a_n\plus{}5}{n\plus{}2}\minus{}2for each

n\in\mathbb N

. Find all

n

such that

a_n\in\mathbb N

.

21

1

Center of W_a,W_b, W_c are collinear.

Let

ABC

be a triangle.

W_a

is a circle with center on

BC

passing through

A

and perpendicular to circumcircle of

ABC

.

W_b,W_c

W_a,W_b,W_c

are collinear.

20

1

Orthocenter, incenter and three other points lie on a circle

Suppose that

M

is an arbitrary point on side

BC

of triangle

ABC

.

B_1,C_1

are points on

AB,AC

such that

MB = MB_1

and

MC = MC_1

. Suppose that

H,I

are orthocenter of triangle

ABC

and incenter of triangle

MB_1C_1

. Prove that

A,B_1,H,I,C_1

lie on a circle.

19

1

Writng a number as a sum of three good numbers

An integer

n

is called a good number if and only if

|n|

is not square of another intger. Find all integers

m

such that they can be written in infinitely many ways as sum of three different good numbers and product of these three numbers is square of an odd number.

18

1

Tetrahedron with circumcircles having the same radius

In tetrahedron

ABCD

, radius four circumcircles of four faces are equal. Prove that AB\equal{}CD, AC\equal{}BD and AD\equal{}BC.

17

1

How to calculate 2^sqrt{2}

A simple calculator is given to you. (It contains 8 digits and only does the operations +,-,*,/, \sqrt{\mbox{}}) How can you find

3^{\sqrt{2}}

with accuracy of 6 digits.

16

1

Points lie on a circle

Segment

AB

is fixed in plane. Find the largest

n

, such that there are

n

points

P_1,P_2,\dots,P_n

in plane that triangles

ABP_i

are similar for

1\leq i\leq n

. Prove that all of

P_i

's lie on a circle.

12

1

Beautiful

There is a lamp in space.(Consider lamp a point) Do there exist finite number of equal sphers in space that the light of the lamp can not go to the infinite?(If a ray crash in a sphere it stops)

11

1

permutation

assume that X is a set of n number.and

0\leq k\leq n

.the maximum number of permutation which acting on

X

st every two of them have at least k component in common,is

a_{n,k}

.and the maximum nuber of permutation st every two of them have at most k component in common,is

b_{n,k}

. a)proeve that :

a_{n,k}\cdot b_{n,k-1}\leq n!

b)assume that p is prime number,determine the exact value of

a_{p,2}

.

9

1

seems easy

Does there exist an infinite set

S

such that for every

a, b \in S

we have a^2 \plus{} b^2 \minus{} ab \mid (ab)^2.

10

1

Easy one

let p be a prime and a and n be natural numbers such that (p^a -1 )/ (p-1) = 2 ^n find the number of natural divisors of na. :)

8

1

Italian TST 2004 - Problem 5

A positive integer

n

is said to be a perfect power if

n=a^b

for some integers

a,b

with

b>1

.

(\text{a})

Find

2004

perfect powers in arithmetic progression.

(\text{b})

Prove that perfect powers cannot form an infinite arithmetic progression.

7

1

A very beautiful problem

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

are polynomials with integer coefficients. Prove there exist a reducible

g(x)

with integer coefficients that

f_{1}+g,f_{2}+g,\dots,f_{n}+g

are irreducible.

6

1

show they are parallel.

let the incircle of a triangle ABC touch BC,AC,AB at A1,B1,C1 respectively. M and N are the midpoints of AB1 and AC1 respectively. MN meets A1C1 at T . draw two tangents TP and TQ through T to incircle. PQ meets MN at L and B1C1 meets PQ at K . assume I is the center of the incircle . prove IK is parallel to AL

13

1

the hardest and the most beautiful in iran(2003)

here is the most difficult and the most beautiful problem occurs in 21th iranian (2003) olympiad assume that P is n-gon ,lying on the plane ,we name its edge 1,2,..,n. if S=s1,s2,s3,.... be a finite or infinite sequence such that for each i, si is in {1,2,...,n}, we move P on the plane according to the S in this form: at first we reflect P through the s1 ( s1 means the edge which iys number is s1)then through s2 and so on like the figure below. a)show that there exist the infinite sequence S sucth that if we move P according to S we cover all the plane b)prove that the sequence in a) isn't periodic. c)assume that P is regular pentagon ,which the radius of its circumcircle is 1,and D is circle ,with radius 1.00001 ,arbitrarily in the plane .does exist a sequence S such that we move P according to S then P reside in D completely?

4

1

constant circle

XOY is angle in the plane.A,B are variable point on OX,OY such that 1/OA+1/OB=1/K (k is constant).draw two circles with diameter OA and OB.prove that common external tangent to these circles is tangent to the constant circle( ditermine the radius and the locus of its center).

5

1

sum of primitive root

Let

p

be an odd prime number. Let

S

be the sum of all primitive roots modulo

p

. Show that if

p-1

isn't squarefree (i. e., if there exist integers

k

and

m

with

k>1

and

p-1=k^2m

), then

S \equiv 0 \mod p

.
If not, then what is

S

congruent to

\mod p

?

14

1

is it hard?

n \geq 6 is an integer. evaluate the minimum of f(n) s.t: any graph with n vertices and f(n) edge contains two cycle which are distinct( also they have no comon vertice)?

3

1

another nice problem!

assume that A is a finite subset of prime numbers, and a is an positive integer. prove that there are only finitely many positive integers m s.t: prime divisors of a^m-1 are contained in A.

2

1

quadrilateral ....

assume ABCD a convex quadrilatral. P and Q are on BC and DC respectively such that angle BAP= angle DAQ .prove that [ADQ]=[ABP] ([ABC] means its area ) iff the line which crosses through the orthocenters of these traingles , is perpendicular to AC.

15

1

nice problem! [variations on the Plotkin bound]

Assume

m\times n

matrix which is filled with just 0, 1 and any two row differ in at least

n/2

members, show that

m \leq 2n

. ( for example the diffrence of this two row is only in one index 110 100)
Edited by Myth

1

does it look easy?

suppose this equation: x ² +y ² +z ² =w ² . show that the solution of this equation ( if w,z have same parity) are in this form: x=2d(XZ-YW), y=2d(XW+YZ),z=d(X ² +Y ² -Z ² -W ² ),w=d(X ² +Y ² +Z ² +W ² )

2003 Iran MO (3rd Round)

Subcontests

Polynomials commuting with z^2+c

Points in space with non-acute angles between them

Square sets

Median has fixed length

DD', EE' and FF' intersect at Q

a fixed point on circle AMN

Homogeneous linear recursive sequences

When does this sequence has integer values?

Center of W_a,W_b, W_c are collinear.

Orthocenter, incenter and three other points lie on a circle

Writng a number as a sum of three good numbers

Tetrahedron with circumcircles having the same radius

How to calculate 2^sqrt{2}

Points lie on a circle

Beautiful

permutation

seems easy

Easy one

Italian TST 2004 - Problem 5

A very beautiful problem

show they are parallel.

the hardest and the most beautiful in iran(2003)

constant circle

sum of primitive root

is it hard?

another nice problem!

quadrilateral ....

nice problem! [variations on the Plotkin bound]

does it look easy?