MathDB

1

n infinite arithmetic progressions of positive integers

P_1,P_2,\ldots ,P_n

, be infinite arithmetic progressions of positive integers, of differences

d_1,d_2,\ldots ,d_n

, respectively. Prove that if every positive integer appears in at least one of the

n

progressions then one of the differences

d_i

divides the least common multiple of the remaining

n-1

differences.
Note:

P_i=\left \{ a_i,a_i+d_i,a_i+2d_i,a_i+3d_i,a_i+4d_i,\cdots \right \}

with

a_i

and

d_i

positive integers.

4

1

after completing 2001 steps, only one number remains on the board.

Initially on the blackboard all the integers from

1

to

2002

inclusive are written in one line and in some order, without repetitions. In each step, the first and second numbers of the line are deleted and the absolute value of the subtraction of the two numbers that have just been deleted is written at the beginning of the line; the other numbers are not modified in that step, and there is a new line that has one less number than the previous step. After completing

2001

steps, only one number remains on the board. Determine all possible values of the number left on the board by varying the order of the

2002

numbers on the initial line (and performing the

2001

steps).

1

sum of numbers of fibanacci like sequence

On the computer screen there are initially two

1

's written. The insert program causes the sum of those numbers to be inserted between each pair of numbers by pressing the

Enter

key. In the first step a number is inserted and we obtain

1-2-1

; In the second step two numbers are inserted and we have

1-3-2-3-1

; In the third, four numbers are inserted and you have

1-4-3-5-2-5-3-4-1

; etc Find the sum of all the numbers that appear on the screen at the end of step number

25

.

2

1

2002x+273y=200201+k diophantine

Determine the smallest positive integer

k

so that the equation

2002x+273y=200201+k

has integer solutions, and for that value of

k

, find the number of solutions

\left (x,y\right )

with

x

,

y

positive integers that have the equation.

3

1

4 circles, tangent in many ways, r_1+r_2+r_3=r

In a circumference

\Gamma

a chord

PQ

is considered such that the segment that joins the midpoint of the smallest arc

PQ

and the midpoint of the segment

PQ

measures

1

. Let

\Gamma_1, \Gamma_2

and

\Gamma_3

be three tangent circumferences to the chord

PQ

that are in the same half plane than the center of

\Gamma

with respect to the line

PQ

. Furthermore,

\Gamma_1

and

\Gamma_3

are internally tangent to

\Gamma

and externally tangent to

\Gamma_2

, and the centers of

\Gamma_1

and

\Gamma_3

are on different halfplanes with respect to the line determined by the centers of

\Gamma

and

\Gamma_2

. If the sum of the radii of

\Gamma_1, \Gamma_2

and

\Gamma_3

is equal to the radius of

\Gamma

, calculate the radius of

\Gamma_2

.

5

1

perpendicularity wanted, isosceles triangle and parallelogram related

Let

\vartriangle ABC

be an isosceles triangle with

AC = BC

. Points

D, E, F

are considered on

BC, CA, AB

, respectively, such that

AF> BF

and that the quadrilateral

CEFD

is a parallelogram. The perpendicular line to

BC

drawn by

B

intersects the perpendicular bisector of

AB

at

G

. Prove that

DE \perp FG

.

2002 Argentina National Olympiad

Subcontests

n infinite arithmetic progressions of positive integers

after completing 2001 steps, only one number remains on the board.

sum of numbers of fibanacci like sequence

2002x+273y=200201+k diophantine

4 circles, tangent in many ways, r_1+r_2+r_3=r

perpendicularity wanted, isosceles triangle and parallelogram related