MathDB

1

3ab < 1 if ax^2 +b=x+1/x has ezctly 2 roots

a

and

b

be real numbers. It is known that the graph of the parabola

y =ax^2 +b

cuts the graph of the curve

y = x+1/x

in exactly three points. Prove that

3ab < 1

.

7

1

(a + 1)(b- 1) = a^2b^2 diophantine

Find all pairs of integers

(a, b)

that satisfy the equation

(a + 1)(b- 1) = a^2b^2.

9

1

cuban concurrency, 3 arc midpoints of circumcircle related

The triangle

ABC

is inscribed in circle

\Gamma

. The points X, Y, Z are the midpoints of the arcs

BC

,

CA

and

AB

respectively in

\Gamma

(those that do not contain the third vertex, in each case). The intersection points of the sides of the triangles

\vartriangle ABC

and

\vartriangle XY Z

form the hexagon

DEFGHK

. Prove that the diagonals

DG

,

EH

and

FK

are concurrent

1

2

numbers 1 - 260 in two 20 x13 grids

We have two

20 \times 13

rectangular grids with

260

unit cells. each one. We insert in the boxes of each of the grids the numbers

1, 2, ..., 260

as follows:

\bullet

For the first grid, we start by inserting the numbers

1, 2, ..., 13

in the boxes in the top row from left to right. We continue inserting numbers

14

,

15

,

...

,

26

in the second row from left to right. We maintain the same procedure until in the last row,

20

, the numbers are placed

248

,

249

,

...

,

260

from left to right.

\bullet

For the second grid we start by inserting the numbers

1

,

2

,

..

.,

20

from top to bottom in the farthest column right. We continue inserting the numbers

21

,

22

,

...

,

40

in the second column from the right also from top to bottom.
We maintain that same procedure until we reach the column on the left where we place the numbers from top to bottom

241

,

242

,

...

,

260

.
Determines the integers inserted in the boxes located in the same position in both grids.

m^4 + 2n^2 = 9mn

Find all the integer solutions of the equation

m^4 + 2n^2 = 9mn

.

5

2

2 player game with (n- divisor of n)

The number 2013 is written on a blackboard. Two players participate, alternating in turns, in the next game. A movement consists in changing the number that is on the board for the difference of this number and one of its divisors. The player who writes a zero loses. Which of the two players can guarantee victory?

x^2 - y = z^2, y^2 - z = x^2, z^2 - x = y^2

Determine all real solutions to the system of equations:

x^2 - y = z^2

y^2 - z = x^2

z^2 - x = y^2

4

2

blue or yellow unit squares in 5x5 board

Each unit square of a

5 \times 5

board is colored blue or yellow. Prove that there is a rectangle with sides parallel to the sides. axes of the board, such that its four corners are the same color.

(a^2b + a) / ( a^2 + b) are integers

Find all positive integers

a, b

such that the numbers

\frac{a^2b + a}{a^2 + b}

and

\frac{ab^2 + b}{b^2 - a}

are integers.

3

2

2 player games with stoines on each vertex of square

Ana and Carlos entertain themselves with the next game. At the beginning of game in each vertex of the square there is an empty box. In each step, the corresponding player has two possibilities: either he adds a stone to an arbitrary box, or move each box clockwise to the next vertex of the square. Carlos starts and they take 2012 steps in turn (each player 1006). So Carlos marks one of the vertices of the square and allows Ana to make a more play. Carlos wins if after this last step the number ofstones in some box is greater than the number of stones in the box which is at the vertex marked by Carlos; otherwise Ana wins. Which of the two players has a winning strategy?

common chord of 2 circles through A 2014 Cuba 2.3

Let

\Gamma_1

and

\Gamma_2

be the circles with diameters

AP

and

AQ

. Let

T

be another point of intersection of the circles

\Gamma_1

and

\Gamma_2

. Let

Q_1

be another point of intersection of the circle

\Gamma_1

and the line

AQ

, and

P_1

the other point of intersection of the circle

\Gamma_2

and the line

AP

. The circle

\Gamma_3

passes through the points

T

,

P

and

P_1

and the circle

\Gamma_4

passes through the points

T

,

Q

and

Q_1

. Prove that the line containing the common chord of the circles

\Gamma_3

and

\Gamma_4

passes through

A

.

2

a / (b+1) + b / (a+1) <= 1

Let

a

and

b

be real numbers with

0 \le a, b \le 1

. (a) Prove that

\frac{a}{b+1} +\frac{b}{a+1} \le 1.

(b) Find the case of equality.

a,b -> |a-b| , starting with 1-2012 in a blackboard

The numbers

1, 2, ..., 2012

are written on a blackboard, in some order, each of them exactly once. Between every two neighboring numbers the absolute value of their difference is written and the original numbers are deleted. This process is repeated until only a number remains on the board. What is the largest number that can stay on the board?

6

1

BC tangent ot (DEF) iff BD=//DC - 2014 Cuba 2.6

Let

ABC

be an isosceles triangle with

AB = AC

. Points

D

,

E

and

F

are on sides

BC

,

CA

and

AB

respectively, such that

\angle FDE =\angle ABC

and

FE

is not parallel to

BC

. Prove that

BC

is tangent to the circumcircle of the triangle

DEF

, if and only if,

D

is the midpoint of

BC

.

2014 Cuba MO

Subcontests

3ab < 1 if ax^2 +b=x+1/x has ezctly 2 roots

(a + 1)(b- 1) = a^2b^2 diophantine

cuban concurrency, 3 arc midpoints of circumcircle related

numbers 1 - 260 in two 20 x13 grids

m^4 + 2n^2 = 9mn

2 player game with (n- divisor of n)

x^2 - y = z^2, y^2 - z = x^2, z^2 - x = y^2

blue or yellow unit squares in 5x5 board

(a^2b + a) / ( a^2 + b) are integers

2 player games with stoines on each vertex of square

common chord of 2 circles through A 2014 Cuba 2.3

a / (b+1) + b / (a+1) <= 1

a,b -> |a-b| , starting with 1-2012 in a blackboard

BC tangent ot (DEF) iff BD=//DC - 2014 Cuba 2.6

2014 Cuba MO

Subcontests

3ab &lt; 1 if ax^2 +b=x+1/x has ezctly 2 roots

(a + 1)(b- 1) = a^2b^2 diophantine

cuban concurrency, 3 arc midpoints of circumcircle related

numbers 1 - 260 in two 20 x13 grids

m^4 + 2n^2 = 9mn

2 player game with (n- divisor of n)

x^2 - y = z^2, y^2 - z = x^2, z^2 - x = y^2

blue or yellow unit squares in 5x5 board

(a^2b + a) / ( a^2 + b) are integers

2 player games with stoines on each vertex of square

common chord of 2 circles through A 2014 Cuba 2.3

a / (b+1) + b / (a+1) &lt;= 1

a,b -&gt; |a-b| , starting with 1-2012 in a blackboard

BC tangent ot (DEF) iff BD=//DC - 2014 Cuba 2.6

3ab < 1 if ax^2 +b=x+1/x has ezctly 2 roots

a / (b+1) + b / (a+1) <= 1

a,b -> |a-b| , starting with 1-2012 in a blackboard