2008 Cuba MO

Part of Cuba MO

Subcontests

(9)

1

1/AM + 1/ AN <= 2/ AB

We have an isosceles triangle

ABC

BC

. Through vertex

A

r

BC

P, Q

are located on the perpendicular bisectors of

AB

AC

respectively, such that

PQ\perp BC

. They are points

M

N

r

\angle APM = \angle AQN = 90^o

\frac{1}{AM} + \frac{1}{AN}\le \frac{2}{ AB}

1

2008 pieces in 2008 x 2008 board

There is a board of

2008\times 2008

2008

pieces, one in each row and each column of the board. It is allowed to do one of the following movements:
a) Take two steps to the right and

10

up. b) Take two steps to the right and

6

steps down. c) Take two steps to the left and

6

steps up. d) Take two steps to the left and

10

steps down.
If the path down cannot be completed, it is skipped to the upper part along the same column and the route continues normally, similarly in the other directions. In each play you will move a checker using any of the allowed operations. Would it be possible that at some point, after a finite number of played, the pieces are located forming a square of side

44

in the upper left corner of the board and the remaining

72

are in the last row in the first

72

1

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

Determine all functions

f : R \to R

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

for all real numbers

x, y

1

a^2+a+b^2>= a^4+a^3+b^4 - 2008 Cuba MO 2.7

For non negative reals

a,b

a^2+a+b^2\ge a^4+a^3+b^4

\frac{1-a^4}{a^2}\ge \frac{b^2-1}{b}

1

locus of centers of rectangles inscribed in acute triangle

ABC

an acute-angle triangle. Let

R

be a rectangle with vertices in the edges of

ABC

O

be the center of

R

. a) Find the locus of all the points

O

. b) Decide if there is a point that is the center of three of these rectangles.

2

three numbers in a blackboard, x-> a+b-1

A boy write three times the natural number

n

in a blackboard. He then performed an operation of the following type several times:
He erased one of the numbers and wrote in its place the sum of the two others minus

1

. After several moves, one of the three numbers in the blackboard is

900

.
Find all the posible values of

n

(m+1)/n +(n+1)/ m is integer

Prove that there are infinitely many ordered pairs of positive integers

(m, n)

\frac{m+1}{n}+\frac{n+1}{m}

is a positive integer.

2

min sum of distances from a point to vertices and edges of regular hexagon

H

a regular hexagon and let

P

a point in the plane of

H

V(P)

the sum of the distances from

P

to the vertices of

H

L(P)

the sum of the distances from

P

to the edges of

H

.
a) Find all points

P

L(P)

is minimun b) Find all points

P

V(P)

APx AC = AM xAB + AN x AD.

Consider the parallelogram

ABCD

. A circle is drawn that passes through

A

and intersects side

AD

N

AB

M

AC

P

such that points

A, M, N, P

are different. Prove that

AP\cdot AC = AM \cdot AB + AN \cdot AD.

2

numbers 1-8 in 9x9 board

We place the numbers from

1

81

9\times

board. Prove that exist

k \in \{1,2,...,9\}

so that the product of the numbers in the

k

-th column is diferent to the product of the numbers in the

k

S(p) = (a -b)^2 + (b - c)^2 + (c - a)^2 >=ra^2

Given a polynomial of degree

2

p(x) = ax^2 +bx+c

define the function

S(p) = (a -b)^2 + (b - c)^2 + (c - a)^2.

Determine the real number

r

such that, for any polynomial

p(x)

2

with real roots, holds

S(p) \ge ra^2

1

Color the plane with two colors

Today was realized the National Olimpiad in Cuba, this is the 3rd problem of the second day: Prof that we can color the lattice points in the plane with two color so that every rectangle with vertices in the lattice points and edges parallels to the co-ordinate axis that have area 2^n is not monocromatic