MathDB

1

Prove that there exists a B when there exists an A

(a_m)

be a sequence satisfying

a_n \geq 0

,

n=0,1,2,\ldots

Suppose there exists

A >0

,

a_m - a_{m+1}

\geq A a_m ^2

for all

m \geq 0

. Prove that there exists

B>0

such that \begin{align*} a_n \le \frac{B}{n} \qquad \qquad \text{for }1 \le n \end{align*}

A5

1

Find f and x so that the equality holds

Consider an integer

n \geq 1

,

a_1,a_2, \ldots , a_n

real numbers in

[-1,1]

satisfying \begin{align*}a_1+a_2+\ldots +a_n=0 \end{align*} and a function

f: [-1,1] \mapsto \mathbb{R}

such \begin{align*} \mid f(x)-f(y) \mid \le \mid x-y \mid \end{align*} for every

x,y \in [-1,1]

. Prove \begin{align*} \left| f(x) - \frac{f(a_1) +f(a_2) + \ldots + f(a_n)}{n} \right| \le 1 \end{align*} for every

x

\in [-1,1]

. For a given sequence

a_1,a_2, \ldots ,a_n

, Find

f

and

x

so hat the equality holds.

A4

1

A verbose problem asking something which the problem itself doesn't know

We consider the set \begin{align*} \mathbb{C}^{\nu} = \{ (z_1,z_2, \ldots , z_{\nu}) \in \mathbb{C} \},\qquad \nu \geq 2 \end{align*} and the function

\phi : \mathbb{C}^{\nu} \longrightarrow \mathbb{C}^{\nu}

mapping every element

(z_1,z_2, \ldots , z_{\nu}) \in \mathbb{C}^{\nu}

to \begin{align*}\phi ( z_1,z_2, \ldots , z_{\nu})= \left( z_1-z_2, z_2-z_3, \ldots, z_{\nu}-z_1 \right) \end{align*} We also consider the

\nu-

tuple

(\omega_0, \omega_1, \ldots , \omega_{\nu-1} )

\in \mathbb{C}^{\nu}

of the

n-

th roots of

-1

, where \begin{align*} \omega_{\mu} = \cos \left( \frac{\pi + 2\mu \pi }{\nu} \right) + \iota \sin \left( \frac{\pi + 2\mu \pi}{\nu} \right) \qquad \mu =0,1, \ldots , \nu -1 \end{align*} Let after

\kappa

(where

\kappa

\in

\mathbb{N}

), successive applications of

\phi

to the element

(\omega_0, \omega_1, \ldots , \omega_{\nu-1} )

, we obtain the element \begin{align*} \phi ^{(\kappa)} \left( \omega_0, \omega_1, \ldots , \omega_{\nu-1} \right) =\left( Z_{\kappa 1}, Z_{\kappa 2}, \ldots , Z_{\kappa \nu } \right) \end{align*} Determine
[*] the values of

\nu

for which all coordinates of

\phi ^{(\kappa)} \left( \omega_0, \omega_1, \ldots , \omega_{\nu-1} \right)

have measures less than or equal to

1

[*] for

\nu =4

, the minimal value of

\kappa \in \mathbb{N}

, for which \begin{align*} \mid Z_{\kappa i} \mid \geq 2^{100} \qquad \qquad 1 \le i \le 4 \end{align*}

A6

1

Inequality on three variables

Prove that if

x,y,z \in \mathbb{R}^+

such that

xy,yz,zx

are sidelengths of a triangle and

k

\in

[-1,1]

, then \begin{align*} \sum \frac{\sqrt{xy}}{\sqrt{xz+yz+kxy}} \geq 2 \sqrt{1-k} \end{align*} Determine the equality condition too.

A7

1

A four variable inequality

Let

x,y,z,t \in \mathbb{R}_{\geq 0}

. Show \begin{align*} \sqrt{xy}+\sqrt{xz}+\sqrt{xt}+\sqrt{yz}+\sqrt{yt}+\sqrt{zt} \geq 3 \sqrt[3]{xyz+xyt+xzt+yzt} \end{align*} and determine the equality cases.

G1

1

m_a^2/r_{bc}+m_b^2/r_{ab}+m_c^2/r_{ab} >= 27\sqrt3 /8 \sqrt[3]{abc}

In acute angled triangle

ABC

we denote by

a,b,c

the side lengths, by

m_a,m_b,m_c

the median lengths and by

r_{b}c,r_{ca},r_{ab}

the radii of the circles tangents to two sides and to circumscribed circle of the triangle, respectively. Prove that

\frac{m_a^2}{r_{bc}}+\frac{m_b^2}{r_{ab}}+\frac{m_c^2}{r_{ab}} \ge \frac{27\sqrt3}{8}\sqrt[3]{abc}

G6

1

collinearity wanted, median, altitudes, 3 circles related

On triangle

ABC

the

AM

(

M\in BC

) is median and

BB_1

and

CC_1

(

B_1 \in AC,C_1 \in AB

) are altitudes. The stright line

d

is perpendicular to

AM

at the point

A

and intersect the lines

BB_1

and

CC_1

at the points

E

and

F

respectively. Let denoted with

\omega

the circle passing through the points

E, M

and

F

and with

\omega_1

and with

\omega_2

the circles that are tangent to segment

EF

and with

\omega

at the arc

EF

which is not contain the point

M

. If the points

P

and

Q

are intersections points for

\omega_1

and

\omega_2

then prove that the points

P, Q

and

M

are collinear.

G7

1

circumcircle contains a point independent of the position of the points X,Y

In the non-isosceles triangle

ABC

consider the points

X

on

[AB]

and

Y

on

[AC]

such that

[BX]=[CY]

,

M

and

N

are the midpoints of the segments

[BC]

, respectively

[XY]

, and the straight lines

XY

and

BC

meet in

K

. Prove that the circumcircle of triangle

KMN

contains a point, different from

M

, which is independent of the position of the points

X

and

Y

.

G8

1

max (AP, BP, CP)>=\sqrt{d_a^2+d_b^2+d_c^2} where d_i distances of interior P

Let

P

be a point in the interior of a triangle

ABC

and let

d_a,d_b,d_c

be its distances to

BC,CA,AB

respectively. Prove that max

(AP, BP, CP) \ge \sqrt{d_a^2+d_b^2+d_c^2}

G5

1

concurrency, circle tangent ext. to circumcicle and 2 extensions of sides

The circle

k_a

touches the extensions of sides

AB

and

BC

, as well as the circumscribed circle of the triangle

ABC

(from the outside). We denote the intersection of

k_a

with the circumscribed circle of the triangle

ABC

by

A'

. Analogously, we define points

B'

and

C'

. Prove that the lines

AA',BB'

and

CC'

intersect in one point.

G4

1

barycenter of intersections of perp. bisectors of GA,GB,GC is the orthocenter

A triangle

ABC

is given with barycentre

G

and circumcentre

O

. The perpendicular bisectors of

GA, GB

meet at

C_1

,of

GB,GC

meet at

A _1

, and

GC,GA

meet at

B_1

. Prove that

O

is the barycenter of the triangle

A_1B_1C_1

.

G3

1

angles wanted, in perimeter and areas bisectors passing through orthocenter

We draw two lines

(\ell_1) , (\ell_2)

through the orthocenter

H

of the triangle

ABC

such that each one is dividing the triangle into two figures of equal area and equal perimeters. Find the angles of the triangle.

A1

1

Inequality for all positive reals

For all

\alpha_1, \alpha_2,\alpha_3 \in \mathbb{R}^+

, Prove \begin{align*} \sum \frac{1}{2\nu \alpha_1 +\alpha_2+\alpha_3} > \frac{2\nu}{2\nu +1} \left( \sum \frac{1}{\nu \alpha_1 + \nu \alpha_2 + \alpha_3} \right) \end{align*} for every positive real number

\nu

N6

1

determine n such that terms of a_n are perfect squares

Let

(x_n)

,

n=1,2, \ldots

be a sequence defined by

x_1=2008

and \begin{align*} x_1 +x_2 + \ldots + x_{n-1} = \left( n^2-1 \right) x_n \qquad ~ ~ ~ \forall n \geq 2 \end{align*} Let the sequence

a_n=x_n + \frac{1}{n} S_n

,

n=1,2,3, \ldots

where

S_n

=

x_1+x_2 +\ldots +x_n

. Determine the values of

n

for which the terms of the sequence

a_n

are perfect squares of an integer.

N1

1

Number starts with a and upon dividing by a ends with a, wowowow big brain

Prove that for every natural number

a

, there exists a natural number that has the number

a

(the sequence of digits that constitute

a

) at its beginning, and which decreases

a

times when

a

is moved from its beginning to it end (any number zeros that appear in the beginning of the number obtained in this way are to be removed).
Example
[*]

a=4

, then

\underline{4}10256= 4 \cdot 10256\underline{4}

[*]

a=46

, then

\underline{46}0100021743857360295716= 46 \cdot 100021743857360295716\underline{46}

C1

1

Professor D of some univeristy of Somewhere

All

n+3

offices of University of Somewhere are numbered with numbers

0,1,2, \ldots ,

n+1,

n+2

for some

n \in \mathbb{N}

. One day, Professor

D

came up with a polynomial with real coefficients and power

n

. Then, on the door of every office he wrote the value of that polynomial evaluated in the number assigned to that office. On the

i

th office, for

i

\in

\{0,1, \ldots, n+1 \}

he wrote

2^i

and on the

(n+2)

th office he wrote

2^{n+2}

-n-3

.
[*] Prove that Professor D made a calculation error [*] Assuming that Professor D made a calculation error, what is the smallest number of errors he made? Prove that in this case the errors are uniquely determined, find them and correct them.

C4

1

A sequence such that pawn returns to original square

An array

n \times n

is given, consisting of

n^2

unit squares. A pawn is placed arbitrarily on a unit square. A move of the pawn means a jump from a square of the

k

th column to any square of the

k

th row. Show that there exists a sequence of

n^2

moves of the pawn so that all the unit squares of the array are visited once and, in the end, the pawn returns to the original position.

C2

1

Cities in countries, maximum no. of flights offered

In one of the countries, there are

n \geq 5

cities operated by two airline companies. Every two cities are operated in both directions by at most one of the companies. The government introduced a restriction that all round trips that a company can offer should have atleast six cities. Prove that there are no more than

\lfloor \tfrac{n^2}{3} \rfloor

flights offered by these companies.

N5

1

Infinitely many n exist such that b divides a_n

Let

(a_n)

be a sequence with

a_1=0

and

a_{n+1}=2+a_n

for odd

n

and

a_{n+1}=2a_n

for even

n

. Prove that for each prime

p>3

, the number \begin{align*} b=\frac{2^{2p}-1}{3} \mid a_n \end{align*} for infinitely many values of

n

N3

1

Proving that terms of sequence are not divisible by 4

The sequence

(\chi_n) _{n=1}^{\infty}

is defined as follows \begin{align*} \chi_{n+1}=\chi_n + \chi _{\lceil \frac{n}{2} \rceil} ~, \chi_1 =1 \end{align*} Prove that none of the terms of this sequence are divisible by

4

N4

1

Diophantine Exponential Equation in primes

Solve the given equation in primes \begin{align*} xyz=1 +2^{y^2+1} \end{align*}

2008 Balkan MO Shortlist

Subcontests

Prove that there exists a B when there exists an A

Find f and x so that the equality holds

A verbose problem asking something which the problem itself doesn't know

Inequality on three variables

A four variable inequality

m_a^2/r_{bc}+m_b^2/r_{ab}+m_c^2/r_{ab} >= 27\sqrt3 /8 \sqrt[3]{abc}

collinearity wanted, median, altitudes, 3 circles related

circumcircle contains a point independent of the position of the points X,Y

max (AP, BP, CP)>=\sqrt{d_a^2+d_b^2+d_c^2} where d_i distances of interior P

concurrency, circle tangent ext. to circumcicle and 2 extensions of sides

barycenter of intersections of perp. bisectors of GA,GB,GC is the orthocenter

angles wanted, in perimeter and areas bisectors passing through orthocenter

Inequality for all positive reals

determine n such that terms of a_n are perfect squares

Number starts with a and upon dividing by a ends with a, wowowow big brain

Professor D of some univeristy of Somewhere

A sequence such that pawn returns to original square

Cities in countries, maximum no. of flights offered

Infinitely many n exist such that b divides a_n

Proving that terms of sequence are not divisible by 4

Diophantine Exponential Equation in primes

2008 Balkan MO Shortlist

Subcontests

Prove that there exists a B when there exists an A

Find f and x so that the equality holds

A verbose problem asking something which the problem itself doesn't know

Inequality on three variables

A four variable inequality

m_a^2/r_{bc}+m_b^2/r_{ab}+m_c^2/r_{ab} &gt;= 27\sqrt3 /8 \sqrt[3]{abc}

collinearity wanted, median, altitudes, 3 circles related

circumcircle contains a point independent of the position of the points X,Y

max (AP, BP, CP)&gt;=\sqrt{d_a^2+d_b^2+d_c^2} where d_i distances of interior P

concurrency, circle tangent ext. to circumcicle and 2 extensions of sides

barycenter of intersections of perp. bisectors of GA,GB,GC is the orthocenter

angles wanted, in perimeter and areas bisectors passing through orthocenter

Inequality for all positive reals

determine n such that terms of a_n are perfect squares

Number starts with a and upon dividing by a ends with a, wowowow big brain

Professor D of some univeristy of Somewhere

A sequence such that pawn returns to original square

Cities in countries, maximum no. of flights offered

Infinitely many n exist such that b divides a_n

Proving that terms of sequence are not divisible by 4

Diophantine Exponential Equation in primes

m_a^2/r_{bc}+m_b^2/r_{ab}+m_c^2/r_{ab} >= 27\sqrt3 /8 \sqrt[3]{abc}

max (AP, BP, CP)>=\sqrt{d_a^2+d_b^2+d_c^2} where d_i distances of interior P