MathDB

1

A series

k

be an even positive integer and define a sequence

<x_n>

by

x_1= 1 , x_{n+1} = k^{x_n} +1.

Show that

x_n ^2

divides

x_{n-1}x_{n+1}

for each

n \geq 2.

26

1

A summation

Let

a_1,a_2,\ldots a_n

be real numbers such that their sum is equal to zero. Find the value of

\sum_{j=1}^{n} \frac{1}{a_j (a_j +a _{j+1}) (a_j + a_{j+1} + a_{j+2}) \ldots (a_j + \ldots a_{j+n-2})}.

where the subscripts are taken modulo

n

assuming none of the denominators is zero.

25

1

A cubic

Find all pairs of cubic equations

x^3 +ax^2 +bx +c =0

and

x^3 +bx^2 + ax +c = 0

where

a,b,c

are integers, such that each equation has three integer roots and both the equations have exactly one common root.

24

1

An equation solutions

Find all nonnegative integers

x,y

such that

2 \cdot 3^{x} +1 = 7 \cdot 5^{y}.

23

1

Another incircle

Let

\Gamma

be the incircle of an equilateral triangle

ABC

of side length

2

units. (a) Show that for all points

P

on

\Gamma

,

PA^2 +PB^2 +PC^2 = 5

. (b) Show that for all points

P

on

\Gamma

, it is possible to construct a triangle of sides equal to

PA,PB,PC

and whose area is equal to

\frac{\sqrt{3}}{4}

units.

22

1

Coloring problem

Consider the points

P

=

(0,0)

,

Q

=

(1,0)

,

R

=

(2,0)

,

S

=

(3,0)

in the

xy

-plane. Let

A,B,C,D

be four finite sets of colours(not necessarily distinct nor disjoint). In how many ways can

P,Q,R

be coloured bu colours in

A,B,C

respectively if adjacent points have to get different colours? In how many ways can

P,Q,R,S

be coloured by colours in

A,B,C,D

respectively if adjacent points have to get different colors?

21

1

A variation of four square theorem

Find all positive integers

n

that can be uniquely expressed as a sum of five or fewer squares.

20

1

Ratios

In the following, the point of intersection of two lines

g

and

h

will be abbreviated as

g\cap h

. Suppose

ABC

is a triangle in which \angle A \equal{} 90^{\circ} and

\angle B > \angle C

. Let

O

be the circumcircle of the triangle

ABC

. Let

l_{A}

and

l_{B}

be the tangents to the circle

O

at

A

and

B

, respectively. Let BC \cap l_{A} \equal{} S and AC \cap l_{B} \equal{} D. Furthermore, let AB \cap DS \equal{} E, and let CE \cap l_{A} \equal{} T. Denote by

P

the foot of the perpendicular from

E

on

l_{A}

. Denote by

Q

the point of intersection of the line

CP

with the circle

O

(different from

C

). Denote by

R

be the point of intersection of the line

QT

with the circle

O

(different from

Q

). Finally, define U \equal{} BR \cap l_{A}. Prove that \frac {SU \cdot SP}{TU \cdot TP} \equal{} \frac {SA^{2}}{TA^{2}}.

19

1

Func eqn

Find all functions

f : \mathbb{R} \mapsto \mathbb{R}

such that

f(xy+f(x)) = xf(y) +f(x)

for all

x,y \in \mathbb{R}

.

15

1

A set

Let

X

be a set with

|X| = n

, and let

X_1 , X_2 ,... X_n

be the

n

subsets eith

|X_j| \geq 2

, for

1 \leq j \leq n

. Suppose for each

2

element subset

Y

of

X

, there is a unique

j

in the set

1,2,3....,n

such that

Y \subset X_j

. Prove that

X_j \cap X_k \not= \Phi

for all

1 \leq j < k \leq n

14

1

A complex polynomial

Let

f(z) = a_m z^n + a_{n-1} z^{n-1} + \cdots + a_1 z + a_0

be a polynomial of degree

n \geq 3

with real coefficients.Suppose all roots of

f(z) =0

lie in the half plane

{\ z \in \mathbb{C} : Re(z) < 0 \}}

. Prove that

a_k a_{k+3} < a_{k+1}a_{k+2}

for

k = 0,1,2,3,.... n-3

13

1

An estimate

Let

a_1 < a_2 < .... < a_n < 2n

ne

n

positive integers such that

a_j

does not divide

a_k

or

j \not= k

. Prove that

a_1 \geq 2^{k}

where

k

is defined by the condition

3^{k} < 2n < 3^{k+1}

and show that it is the best estimate for

a_1

12

1

Ratio of radii

Let

ABC

be a triangle with vertices at lattice points. Suppose one of its sides in

\sqrt{n}

, where

n

is square-free. Prove that

\frac{R}{r}

is irraational . The symbols have usual meanings.

11

1

Partitioning

(a) Prove that the set

X = (1,2,....100)

cannot be partitoned into THREE subsets such that two numbers differing by a square belong to different subsets. (b) Prove that

X

can so be partitioned into

5

subsets.

10

1

Find an angle

On the sides

AB

and

BC

of triangle

ABC

, points

K

and

M

are chosen such that the quadrilaterals

AKMC

and

KBMN

are cyclic , where

N = AM \cap CK

. If these quads have the same circumradii, find

\angle ABC

9

1

Distribution of balls

In how many ways can

n

identical balls be distributed to nine persons

A,B,C,D,E,F,G,H,I

so that the number of balls recieved by

A

is the same as the total number of balls recieved by

B,C,D,E

together,.

8

1

A num ineq

Prove that For all positive integers

m

and

n

, one has

| n \sqrt{2005} - m | > \frac{1}{90n}

7

1

A num eqn

Fins all ordered triples

\left(a,b,c\right)

of positive integers such that abc \plus{} ab \plus{} c \equal{} a^3.

5

1

Find the triangles

Characterize all triangles

ABC

s.t.

AI_a : BI_b : CI_c = BC: CA : AB

where

I_a

etc. are the corresponding excentres to the vertices

A, B , C

4

1

One on gcds

Let

m,n

be natural numbers and let

d = gcd(m,n)

. Let

x = 2^{m} -1

and

y= 2^n +1

(a) If

\frac{m}{d}

is odd, prove that

gcd(x,y) = 1

(b) If

\frac{m}{d}

is even, Find

gcd(x,y)

3

1

Solve an eqn

Find all real

\alpha

s.t.

[ \sqrt{n + \alpha} + \sqrt{n} ] = [ \sqrt{4n+1} ]

holds for all natural numbers

n

2

1

A sequence of circles

Let

< \Gamma _j >

be a sequnce of concentric circles such that the sequence

< R_j >

, where

R_j

denotes the radius of

\Gamma_j

, is increasing and

R_j \longrightarrow \infty

as

j \longrightarrow \infty

. Let

A_1 B_1 C_1

be a triangle inscribed in

\Gamma _1

. extend the rays

\vec{A_i B_1} , \vec{B_1 C_1 }, \vec{C_1 A_1}

to meet

\Gamma_2

in

B_2, C_2

and

A_2

respectively and form the triangle

A_2 B_2 C_2

. Continue this process. Show that the sequence of triangles

< A_n B_n C_n >

tends to an equilateral triangle as

n \longrightarrow \infty

1

Prove that the sequence..

Consider the sequence

<{a_n}>

of natural numbers such that {i}

a_n

is a square numver for all

n

; (ii)

a_{n+1} - a_n

is either a prime or a square of a prime for each

n

. Show that

<a_n>

is a finite sequence. Determine the longest such sequence.

16

1

Cyclic quad

The diagonals AC and BD of a cyclic ABCD intersect at E. Let O be circumcentre of ABCD. If midpoints of AB, CD, OE are collinear prove that AD=BC. Bomb [Moderator edit: The problem is wrong. See also http://www.mathlinks.ro/Forum/viewtopic.php?t=53090 .]

18

1

(2^a-2^b)/(2^c-2^d)

Find the least positive integer, which may not be represented as

{2^a-2^b\over 2^c-2^d}

, where

a,\,b,\,c,\,d

are positive integers.

17

1

excircles touch corresponding sides of triangle ABC

Let

A',\,B',\,C'

be points, in which excircles touch corresponding sides of triangle

ABC

. Circumcircles of triangles

A'B'C,\,AB'C',\,A'BC'

intersect a circumcircle of

ABC

in points

C_1\ne C,\,A_1\ne A,\,B_1\ne B

respectively. Prove that a triangle

A_1B_1C_1

ABC

touches its sides.

6

1

trapezoid, prove two equations

Let

ABCD

be a trapezoid such that

AB

is parallel to

CD

, and let

E

be the midpoint of its side

BC

. Suppose we can inscribe a circle into the quadrilateral

ABED

, and that we can inscribe a circle into the quadrilateral

AECD

. Denote

|AB|=a

,

|BC|=b

,

|CD|=c

,

|DA|=d

. Prove that

a+c=\frac{b}{3}+d;

\frac{1}{a}+\frac{1}{c}=\frac{3}{b}

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Subcontests

A series

A summation

A cubic

An equation solutions

Another incircle

Coloring problem

A variation of four square theorem

Ratios

Func eqn

A set

A complex polynomial

An estimate

Ratio of radii

Partitioning

Find an angle

Distribution of balls

A num ineq

A num eqn

Find the triangles

One on gcds

Solve an eqn

A sequence of circles

Prove that the sequence..

Cyclic quad

(2^a-2^b)/(2^c-2^d)

excircles touch corresponding sides of triangle ABC

trapezoid, prove two equations