MathDB

1

steps of escalator in shopping mall (Chile NMO 2000 P1)

Professor David proposed to his wife to calculate the steps of an escalator that worked in a shopping mall, asking him to walk up counting the steps that rise from the bottom to the end. The teacher, in turn, left with his wife, but walking the twice as fast, so that the woman advanced one step each time her husband advanced

2

. When The lady arrived at the top reported that she had counted

21

steps, while the teacher counted

28

of them. How many steps are there in sight on the ladder at any given time?
[hide=original wording]El profesor David propuso a su senora calcular los escalones de una escalera mecanica que funcionaba en un centro comercial, pidiendole que caminara hacia arriba contando los escalones que subiera desde la base hasta el final. El profesor a su vez, partio junto a su senora, pero caminando el doble de rapido, de modo que la senora avanzaba un escalon cada vez que su marido avanzaba 2. Cuando la senora llego arriba informo que habıa contado 21 escalones, mientras que el profesor conto 28 de ellos, ¿Cuantos escalones hay a la vista en la escalera en un instante cualquiera?

7

1

a^2 + b^2 not prime, ax (x^2 + ax + 1) = b (x^2 + b + 1) (Chile NMO 2000 P7)

Consider the following equation in

x

:

ax (x^2 + ax + 1) = b (x^2 + b + 1).

It is known that

a, b

are real such that

ab <0

and furthermore the equation has exactly two integer roots positive. Prove that under these conditions

a^2 + b^2

is not a prime number.

6

1

76 3-color tiles in 4x19 rectangle (Chile NMO 2000 P6)

With

76

tiles, of which some are white, other blue and the remaining red, they form a rectangle of

4 \times 19

. Show that there is a rectangle, inside the largest, that has its vertices of the same color.

5

1

m_1m_2...m_n divisible by 2^n (Chile NMO 2000 P5)

Let

n

be a positive number. Prove that there exists an integer

N =\overline{m_1m_2...m_n}

with

m_i \in \{1, 2\}

which is divisible by

2^n

.

3

1

7 divides infinite periodic numbers (Chile NMO 2000 P3)

A number

N_k

is defined as periodic if it is composed in number base

N

of a repeated

k

times . Prove that

7

divides to infinite periodic numbers of the set

N_1, N_2, N_3,...

2

1

interior not in a closed polygon (Chile NMO 2000 P2)

In the plane, we have any polygon that does not intersect itself and is closed. Given a point that is not on the edge of the polygon. How can we determine whether it is inside or outside the polygon? (the polygon has a finite number of sides)
[hide=original wording]En el plano se tiene un poligono cualquiera que no se corta a si mismo y que es cerrado. Dado un punto que no esta sobre el borde del poligono, Como determinara se esta dentro o fuera del poligono? (el poligono tiene un numero nito de lados)

4

1

find the angles of a triangle, sums of lengths given Chile 2000 L2 p4

Let

AD

be the bisector of a triangle

ABC

(D \in BC)

such that

AB + AD = CD

and

AC + AD = BC

. Determine the measure of the angles of

\vartriangle ABC

2000 Chile National Olympiad

Subcontests

steps of escalator in shopping mall (Chile NMO 2000 P1)

a^2 + b^2 not prime, ax (x^2 + ax + 1) = b (x^2 + b + 1) (Chile NMO 2000 P7)

76 3-color tiles in 4x19 rectangle (Chile NMO 2000 P6)

m_1m_2...m_n divisible by 2^n (Chile NMO 2000 P5)

7 divides infinite periodic numbers (Chile NMO 2000 P3)

interior not in a closed polygon (Chile NMO 2000 P2)

find the angles of a triangle, sums of lengths given Chile 2000 L2 p4