MathDB

1

computer divides with 18

A computer program generates a sequence of numbers with the following rule: the first number is written by Camilo; thereafter, the program calculates the integer division of the last number generated by

18

; thus obtains a quotient and a remainder. The sum of that quotient plus that remainder is the next number generated. For example, if Camilo's number is

5291,

the computer makes

5291 = 293 \times 18 + 17

, and generates

310 = 293 + 17

. The next number generated will be

21

, since

310 = 17 \times 18 + 4

and

17 + 4= 21

; etc Whatever Camilo's initial number is, from some point on, the computer always generates the same number. Determine what is that number that will be repeated indefinitely, if Camilo's initial number is equal to

2^{110}.

4

1

4620 is multiple of a, 4620 is multiple of b and b is multiple of a

Determine the number of pairs of natural numbers

(a,b)

that simultaneously verify that

4620

is a multiple of

a

,

4620

is a multiple of

b

and

b

is a multiple of

a

.

3

1

1 or -1 in 32x10 board

There is a board with 32 rows and 10 columns. Pablo writes 1 or -1 in each box. Matías, with Pablo's board in view, chooses one or more columns, and in each of the chosen columns, changes all of Pablo's numbers to their opposites (where there is 1 he puts -1 and where there is -1 he puts 1) . In the other columns, leave Pablo's numbers. Matías wins if he manages to make his board have each of the rows different from all the rows on Pablo's board. Otherwise, that is, if any row on Matías's board is equal to any row on Pablo's board, Pablo wins. If both play perfectly, determine which of the two is assured of victory.

1

sequence of digits by naturals

The natural numbers are written in succession, forming a sequence of digits

12345678910111213141516171819202122232425262728293031\ldots

Determine how many digits the natural number has that contributes to this sequence with the digit in position

10^{2000}

. Clarification: The natural number that contributes to the sequence with the digit in position

10

has

2

digits, because it is

10

; The natural number that contributes to the sequence with the digit at position

10^2

has

2

digits, because it is

55

.

6

1

min area of a quadrilateral folding an equilateral triangle

You have an equilateral paper triangle of area

9

and fold it in two, following a straight line that passes through the center of the triangle and does not contain any vertex of the triangle. Thus there remains a quadrilateral in which the two pieces overlap, and three triangles without overlaps. Determine the smallest possible value of the quadrilateral area of the overlay.

2

1

perpendicular wanted, midpoint, angle bisector, parallel, median related

Given a triangle

ABC

with side

AB

greater than

BC

, let

M

be the midpoint of

AC

and

L

be the point at which the bisector of angle

\angle B

intersects side

AC

. The line parallel to

AB

, which intersects the bisector

BL

at

D

, is drawn by

M

, and the line parallel to the side

BC

that intersects the median

BM

at

E

is drawn by

L

. Show that

ED

is perpendicular to

BL

.

2000 Argentina National Olympiad

Subcontests

computer divides with 18

4620 is multiple of a, 4620 is multiple of b and b is multiple of a

1 or -1 in 32x10 board

sequence of digits by naturals

min area of a quadrilateral folding an equilateral triangle

perpendicular wanted, midpoint, angle bisector, parallel, median related