MathDB

1

sum x_k(2x_k - x_{k+1} - x_{k+2}) / ( x_{k+1} + x_{k+2} ) >= 0

x_1, x_2, ..., x_n

be positive reals. Prove that

\sum_{k=1}^n \frac{x_k(2x_k - x_{k+1} - x_{k+2})}{x_{k+1} + x_{k+2}} \ge 0

In the sum, cyclic indices have been taken, that is,

x_{n+1} = x_1

and

x_{n+2} = x_2

.

9

1

p | 2qr + r, q | 2pr + p y r | 2pq + q.

Find all the triples of prime numbers

(p, q, r)

such that

p | 2qr + r \,\,\,, \,\,\,q |2pr + p \,\,\, and \,\,\, r | 2pq + q.

5

1

(5^n-1)/(n+2) is integer

Prove that there are infinitely many positive integers

n

such that

\frac{5^n-1}{n+2}

is an integer.

6

1

collinear, intersecting circles related

Let

\omega_1

and

\omega_2

be circles that intersect at points

A

and

B

and let

O_1

and

O_2

be their respective centers. We take

M

in

\omega_1

and

N

in

\omega_2

on the same side as

B

with respect to segment

O_1O_2

, such that

MO_1\parallel BO_2

and

BO_1 \parallel NO_2

. Draw the tangents to

\omega_1

and

\omega_2

through

M

and

N

respectively, which intersect at

K

. Show that

A

,

B

and

K

are collinear.

8

1

80-80-20 classic problem, AD = BC

Let

ABC

be an isosceles triangle with base

BC

and

\angle BAC = 20^o

. Let

D

a point on side

AB

such that

AD = BC

. Determine

\angle DCA

.

4

1

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

Determine all the functions

f : R \to R

such that:

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

for all

x, y \in R

.

1

2

routes in 4x7 grid

Juan and Pedro play alternately on the given grid. Each one in turn traces

1

to

5

routes different from the ones outlined above, that join

A

with

B

, moving only to the right and upwards on the grid lines. Juan starts playing. The one who traces a route that passes through

C

or

D

loses. Prove that one of them can win regardless of how the other plays. https://cdn.artofproblemsolving.com/attachments/2/7/6a24ca9c4c1c710bd41e44bfcab3d3b61b6d4f.png

prime mod 30, 60, 90

Show that when a prime number is divided by

30

, the remainder is

1

or a prime number. Shows that if it is divided by

60

or

90

the same thing does not happen.

3

2

Inca nxn board

In each square of an

n \times n

board with

n\ge 2

, an integer is written not null. Said board is called Inca if for each square, the number written on it is equal to the difference of the numbers written on two of its neighboring squares (with a common side). For what values of

n

, can you get Inca boards ?

min x^2 + y^2 + z^2 if x^3 + y^3 + z^3 -3xyz = 1

Determine the smallest value of

x^2 + y^2 + z^2

, where

x, y, z

are real numbers, so that

x^3 + y^3 + z^3 -3xyz = 1.

2

2008^2 hydras in the Hidro planet

In Hidro planet were living

2008^2

hydras some time ago. One of them had 1 tentacle, other 2 and so on to the last with

2008^2

tentacles. Let

t(H)

be the number of tentacles of hydra

H

. The pairing of

H_1

and

H_2

(where

t(H_1) < t(H_2)

) is a new hydra with

t(H_2)-t(H_1)+8

tentacles, in case of

t(H_1)=t(H_2)

they die. An expedition found in Hidro the last hydra with 23 tentacles. That could be true ?

ABC is equilateral if circles (XYZ) and (ABC) are equal

Let

I

be the incenter of an acute riangle

ABC

. Let

C_A(A, AI)

be the circle with center

A

and radius

AI

. Circles

C_B(B, BI)

,

C_C(C, CI)

are defined in an analogous way. Let

X, Y, Z

be the intersection points of

C_B

with

C_C

,

C_C

with

C_A

,

C_A

with

C_B

respectively (different than

I

) . Show that if the radius of the circle that passes through the points

X, Y, Z

is equal to the radius of the circle that passes through points

A

,

B

and

C

then triangle

ABC

is equilateral.

2009 Cuba MO

Subcontests

sum x_k(2x_k - x_{k+1} - x_{k+2}) / ( x_{k+1} + x_{k+2} ) >= 0

p | 2qr + r, q | 2pr + p y r | 2pq + q.

(5^n-1)/(n+2) is integer

collinear, intersecting circles related

80-80-20 classic problem, AD = BC

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

routes in 4x7 grid

prime mod 30, 60, 90

Inca nxn board

min x^2 + y^2 + z^2 if x^3 + y^3 + z^3 -3xyz = 1

2008^2 hydras in the Hidro planet

ABC is equilateral if circles (XYZ) and (ABC) are equal

2009 Cuba MO

Subcontests

sum x_k(2x_k - x_{k+1} - x_{k+2}) / ( x_{k+1} + x_{k+2} ) &gt;= 0

p | 2qr + r, q | 2pr + p y r | 2pq + q.

(5^n-1)/(n+2) is integer

collinear, intersecting circles related

80-80-20 classic problem, AD = BC

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

routes in 4x7 grid

prime mod 30, 60, 90

Inca nxn board

min x^2 + y^2 + z^2 if x^3 + y^3 + z^3 -3xyz = 1

2008^2 hydras in the Hidro planet

ABC is equilateral if circles (XYZ) and (ABC) are equal

sum x_k(2x_k - x_{k+1} - x_{k+2}) / ( x_{k+1} + x_{k+2} ) >= 0