IMO-1999 ( Não é a versão oficial)

[ponce@xxxxxxxxxx (Luiz Ponce )]

Olá amigos,
Eu recebi8 de uma lista da Argentina a IMO-1999
( não é a versão oficial) mas já dá para termos uma
ideia.
1. Find all sets S of points in the plane so that for every to members A
and
B of S the perpendicular bisector of AB is an axis of symmetry for S.
(Originally it was supposed to be the same problem in 3-Dimention)

2. Let n be a fixed Natural number.
For this n find the smallest constant C so that the inequality

Sum(1<=i<j<=n, x_i*x_j*((x_i)^2+(x_j)^2))<=C*(Sum(1<=i<=n, x_i))^4

holds for all real x_i >= 0
Determine when there is equality

3. Let n be a natural even number. Consider an n x n table and mark N
squares. Two squares are adjacent if they have a common side. Find the
smallest N so that every square (marked or unmaked) has an adjacent
marked
square.

4. Determine all pares (n,p) of natural numbers so that:
p is a prime
n<=2p
(p-1)^n + 1 is divisible by n^(p-1)

5. We have two circles G1 and G2 so that G2 passes through the center of
G1.
Concider circle G which touches both G1 and G2 (G1 at N, G2 at M). Let
the
line through the intersetion of G1 and G2 intersect G at A and B and let
AM
intersect G2 at C and BM intersect G2 at D. Prove that CD is a tangent
line
to G1.

6. Find all functions real to real so that
   f(x-f(y)) = f(f(y)) + f(x) + xf(y) - 1
(I am not completly sure that the sixth problem is correct)

Um abraço
PONCE