MathDB

1

no of cars that pass along the highway per hour, from a bus

On a long enough highway, a passenger in a bus observes the traffic. He notes that, during an hour, the bus going with a constant velocity overpasses

a

cars and gets overpassed by

b

cars, while

c

cars pass in the opposite direction. Assuming that the traffic is the same in both directions, is it possible to determine the number of cars that pass along the highway per hour? (You may assume that the velocity of a car can take only two values.)

6

1

(\sqrt2-1)^n=\sqrt{m}-\sqrt{m-1} , for any n exists m>1

Show that for any positive integer

n

there exists an integer

m > 1

such that

(\sqrt2-1)^n=\sqrt{m}-\sqrt{m-1}

.

2

1

a_{n+1} =\frac{a_n}{1+a_n} , a_0 =\alpha > 0 , a_n wanted

Determine the general term of the sequence (

a_n

) given by

a_0 =\alpha > 0

and

a_{n+1} =\frac{a_n}{1+a_n}

.

1

intersecting circles given, segment parallel to given tangent wanted

Two circles

\alpha

and

\beta

intersect at points

P

and

Q

. The lines connecting a point

R

on

\beta

with

P

and

Q

intersect

\alpha

again at

S

and

T

respectively. Prove that

ST

is parallel to the line tangent to

\beta

at

R

.

5

1

tetrahedra with base T whose all faces are acute triangles of the same area

Given an acute triangle

T

with sides

a,b,c

, find the tetrahedra with base

T

whose all faces are acute triangles of the same area.

3

1

Probability

Two numbers are randomly selected from interval

I = [0, 1]

. Given

\alpha \in I

, what is the probability that the smaller of the two numbers does not exceed

\alpha

? Is the answer

(100 \alpha)

%, it just seems too easy. :|

4

1

Point with integer coordinates

Prove that a circle centered at point

(\sqrt{2},\sqrt{3})

in the cartesian plane passes through at most one point with integer coordinates. I tried to prove that any circle with center at

(0,0)

has at most one point with coordinates

(a-\sqrt{2},b-\sqrt{3})

;

a,b \in \mathbb{Z}

. So that when we translate the center to

(\sqrt{2},\sqrt{3})

we have what we wanted to show.

1986 ITAMO

Subcontests

no of cars that pass along the highway per hour, from a bus

(\sqrt2-1)^n=\sqrt{m}-\sqrt{m-1} , for any n exists m>1

a_{n+1} =\frac{a_n}{1+a_n} , a_0 =\alpha > 0 , a_n wanted

intersecting circles given, segment parallel to given tangent wanted

tetrahedra with base T whose all faces are acute triangles of the same area

Probability

Point with integer coordinates

1986 ITAMO

Subcontests

no of cars that pass along the highway per hour, from a bus

(\sqrt2-1)^n=\sqrt{m}-\sqrt{m-1} , for any n exists m&gt;1

a_{n+1} =\frac{a_n}{1+a_n} , a_0 =\alpha &gt; 0 , a_n wanted

intersecting circles given, segment parallel to given tangent wanted

tetrahedra with base T whose all faces are acute triangles of the same area

Probability

Point with integer coordinates

(\sqrt2-1)^n=\sqrt{m}-\sqrt{m-1} , for any n exists m>1

a_{n+1} =\frac{a_n}{1+a_n} , a_0 =\alpha > 0 , a_n wanted