2

Part of 2002 IberoAmerican

Problems(2)

17th ibmo - el salvador 2002./q2

Source: Spanish Communities

Given any set of

9

points in the plane such that there is no

3

of them collinear, show that for each point

P

of the set, the number of triangles with its vertices on the other

8

points and that contain

P

on its interior is even.

modular arithmeticinvariantgeometrygeometric transformationreflectioncalculusintegration

17th ibmo - el salvador 2002./q5.

Source: Spanish Communities

The sequence of real numbers

a_1,a_2,\dots

is defined as follows:

a_1=56

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

n\ge 1

. Show that there is an integer

1\leq{k}\leq2002

a_k<0

inductionalgebra unsolvedalgebra