2012 Cuba MO

Part of Cuba MO

Subcontests

(9)

1

m^2 + n^2 =(m + 1)(n + 1)

(m, n)

of positive integers such that

m^2 + n^2 =(m + 1)(n + 1).

1

<BAO = 30^o, <CBO = 20^o and <ABO =<ACO = 40^o

O

be a point interior to triangle

ABC

\angle BAO = 30^o

\angle CBO = 20^o

\angle ABO = \angle ACO = 40^o

. Knowing that triangle

ABC

is not equilateral, find the measures of its interior angles.

1

(a^2 + b^2)(c^2 + d^2) = (ab + cd)^2, (a^2 + d^2)(b^2 + c^2) = (ad + bc)^2

If the natural numbers

a, b, c, d

verify the relationships:

(a^2 + b^2)(c^2 + d^2) = (ab + cd)^2

(a^2 + d^2)(b^2 + c^2) = (ad + bc)^2

gcd(a, b, c, d) = 1

a + b + c + d

is a perfect square.

1

f(x^2 + f(y)) = y - x^2

Find all the functions

f : R\to R

f(x^2 + f(y)) = y - x^2

x, y

1

AD x BC = AB x DC + BD x AN - 2012 Cuba MO 2.6

ABC

be a right triangle at

A

AD

be the relative height to the hypotenuse. Let

N

be the intersection of the bisector of the angle of vertex

C

AD

AD \cdot BC = AB \cdot DC + BD \cdot AN.

2

a rectangle is formed of 3 × 7 by 21 bw pieces, vertices of rectangle

21

pieces, some white and some black, a rectangle is formed of

3 \times 7

. Prove that there are always four pieces of the same color located at the vertices of a rectangle.

sum xz / (x^2 + xy + y^2 + 6z^2) <= 1/3

x, y, z

be positive reals. Prove that

\frac{xz}{x^2 + xy + y^2 + 6z^2} + \frac{zx}{z^2 + zy + y^2 + 6x^2} + \frac{xy}{x^2 + xz + z^2 + 6y^2} \le \frac13

2

red and blue squares in 123 x 123 board

123 \times 123

board, each square is painted red or blue according to the following conditions:
a) Each square painted red that is not on the edge of the board has exactly

5

blue squares among its

8

neighboring squares.
b) Each square painted blue that is not on the edge of the board has exactly

4

red squares among its

8

neighboring squares.
Determine the number of red-painted squares on the board.

x^2+mx * n = 0

A mathematics teacher writes a quadratic equation on the blackboard of the form

x^2+mx \star n = 0

m

n

integers. The sign of

n

is blurred. Even so, Claudia solves it and obtains integer solutions, one of which is

2011

. Find all possible values of

m

n

2

in a school with 5 different grades there are 250 girls and 250 boys

In a school with 5 different grades there are 250 girls and 250 boys. Each grade has the same number of students. for a competition of knowledge wants to form teams of a female and a male who are of the same grade. If in each grade there are at least

19

19

males. Find the greatest amount of teams that can be formed.

CP//BQ , midpoints, 2012 Cuba MO 2.2

Given the triangle

ABC

L

M

N

be the midpoints of

BC

CA

AB

respectively. The lines

LM

LN

cut the tangent to the circumcircle at

A

P

Q

respectively . Prove that

CP \parallel BQ

2

x_{1006} =? x_1+x_2+...+x_{1006} = 503^2

\frac{x_1}{x_1+1} = \frac{x_2}{x_2+3} = \frac{x_3}{x_3+5} = ...= \frac{x_{1006}}{x_{1006}+2011}

x_1+x_2+...+x_{1006} = 503^2

, determine the value of

x_{1006}

1000 balls of dough 0.38 and 5000 balls of dough 0.038 in boxes

1000

0.38

5000

0.038

that must be packed in boxes. A box contains a collection of balls whose total mass is at most

1

. Find the smallest number of boxes that they are needed.