MathDB

1

cover a unit circle with finite circles (Chile NMO 2001 P2)

Prove that the only way to cover a square of side

1

with a finite number of circles that do not overlap, it is with only one circle of radius greater than or equal to

\frac{1}{\sqrt2}

. Circles can occupy part of the outside of the square and be of different radii.

1

no of integer triangles with sides among 50 and 100 (Chile NMO 2001 P1)

\bullet

In how many ways can triangles be formed whose sides are integers greater than

50

and less than

100

?

\bullet

In how many of these triangles is the perimeter divisible by

3

?

7

1

petrol stations around a circular circuit (Chile NMO 2001 P7)

In a circular circuit there are petrol stations, so that the total accumulated petrol in them it is exactly enough for a car to go around the circuit. Prove that there is a position from where a car, with the tank of finite capacity and initially empty, can leave and get to go a full loop around the circuit, stopping to refuel at positions.
[hide=original wording]En un circuito circular hay puestos de gasolina, de modo que el total de la gasolina acumulada en ellos es exactamente suciente para que un auto de una vuelta completa al circuito. Demostrar que existe un puesto desde donde un auto, con el estanque de capacidad finita e inicialmente vacio, puede partir y conseguir recorrer una vuelta completa al circuito, deteniendose a reabastecerse de gasolina en los puestos.

5

1

finite number of scissor cuts on paper, separate A and B (Chile NMO 2001 P5)

On a right triangle of paper, two points

A

and

B

have been painted. You have scissors and you have the right to make cuts (on paper) as follows: cut through a height of the given triangle. In doing so, remove, without the respective altitude, one of the two triangles and continue the process. Prove that after a finite number of cuts you can separate points

A

and

B

leaving one of them outside the remaining triangles.

4

1

2^{2n}-1 is a multiple of 3 (Chile NMO 2001 P4)

Given a natural number

n

, prove that

2^{2n}-1

is a multiple of

3

.

3

1

3/2} \le /d_a/h_a + d_b/h_b/ + d_c/ h_c, angle bisectors Chile 2001 L2 P3

In a triangle

\vartriangle ABC

, let

h_a, h_b

and

h_c

the atlitudes. Let

D

be the point where the inner bisector of

\angle BAC

cuts to the side

BC

and

d_a

is the distance from the

D

point next to

AB

. The distances

d_b

and

d_c

are similarly defined. Show that:

\dfrac {3} {2} \le \dfrac {d_a} {h_a} + \dfrac {d_b} {h_b} + \dfrac {d_c} {h_c}

For what kind of triangles does the equality hold?

6

1

tangential quadrilateral, starting with 2 equal circles (Chile 2001 L2 P6)

Let

C_1, C_2

be two circles of equal radius, disjoint, of centers

O_1, O_2

, such that

C_1

is to the left of

C_2

. Let

l

be a line parallel to the line

O_1O_2

, secant to both circles. Let

P_1

be a point of

l

, to the left of

C_1

and

P_2

a point of

l

, to the right of

C_2

such that the tangents of

P_1

to

C_1

and of

P_2

a

C_2

form a quadrilateral. Show that there is a circle tangent to the four sides of said quadrilateral.

2001 Chile National Olympiad

Subcontests

cover a unit circle with finite circles (Chile NMO 2001 P2)

no of integer triangles with sides among 50 and 100 (Chile NMO 2001 P1)

petrol stations around a circular circuit (Chile NMO 2001 P7)

finite number of scissor cuts on paper, separate A and B (Chile NMO 2001 P5)

2^{2n}-1 is a multiple of 3 (Chile NMO 2001 P4)

3/2} \le /d_a/h_a + d_b/h_b/ + d_c/ h_c, angle bisectors Chile 2001 L2 P3

tangential quadrilateral, starting with 2 equal circles (Chile 2001 L2 P6)