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[obm-l] Resquício da pequena lista de dúvidas



Olá,
Agradeço aos colegas que ajudaram-me na resolução de algumas questões. Retorno com o resquício das dúvidas.
 
1°) (IMO/1996) Let ABCDEF be a convex hexagon such that AB||DB, BC||///ef and CD||AF. Let Ra, Rc, Re denote the circumradii or triangles FAB, BCD, DEF, respectively, and let P denote the perimeter or hexagon. Prova that Ra + Rc + Re >= P/2.

2°) Vietnam

i) Solve the system of equations

raiz(3x)*[1+1/(x+y)]=2

riaz(7y)*[1-1/(x+y)]=4*raiz(2)

ii) Determine all functions f: N -> N satisfying (for all n E N) f(n) + f(n+1) = f(n+2)*f(n+3) - 1996

iii) Let a,b,c,d be four nonnegative real numbers satisfying  the condition 2(ab + ac + ad + bc+ bd + cd) + abc + abd + acd + bcd = 16. Prove that a + b + c + d >= (2/3)*(ab + ac + ad + bc + bd + cd) and determine when equality occurs.

3°) Irish Mathematics Olympiad Training 2000

i)Find all integers x,y satisfying the equation
x^3+y^3=(x-y).1997^2

ii)Find a solution of the simultaneous equations
3[x-y(x²-y²)^½] = (1-x²+y²)^½
5[y-x(x²-y²)^½] = (1-x²+y²)^½

4°) (Ibero 96) Determine todos os possíveis valores da soma dos dígitos de um quadrado perfeito.