Inequalities, Polynomials and Trigonometric Functions

1.Solve the following system of equtions

x1(x1 - 1) = x2 - 1, x2(x2 - 1) = x3 - 1, ... xn(xn - 1) = x1 - 1.
where xi's are real numbers.

2.Find all such functions f : R -> R that satisfy

xf(x) + f(1-x) = x3 - x .
for all x \in R

3.Prove that if a,x,y,z are real numbers such that

(cos x + cos y + cos z)/cos(x + y + z)= (sin x + sin y + sin z)/sin(x + y + z) = a .
then the following equation holds
cos(x+y) + cos(z+x) + cos(y+z) = a .

4.Let f1, f2, f3,...be the elements of the {Fibonacci sequence}(i.e., f1 = f2 = 1 a  fn+2 = fn+1 + fn for n \in N). Prove that if the polynomial P(x) of degree 998 satisfies P(k) = fk for k = 1000, 1001,..., 1998 , then P(1999) = f1999 - 1.

5.Given n \in N find the minimum value of the expression

x1 + (x22)/2 + (x33)/3 + ... + (xnn)/n ,.
where x1,x2,...,xnare positive real numbers satisfying
1/x1 + 1/x2 + ... + 1/xn = n ,.

Problems were choosen by Juraj Foldes (1, 2, 3, 5), Eugen Kovac (2, 4), Vladimir Marko (2)

Literature

Loren C. Larson, Problem solving through problems, Springer-Verlag,New York, 1983.
J.\ Sedivy: Shodna zobrazeni v~konstruktivnich ulohach, SMM 3 [in Czech]
M.\ Sisler, J.\ Jarnik: O~funkcich, SMM 4 [in Czech]
J.\ Hornik: Ulohy o~maximech a~minimech funcki, SMM 17 [in Czech]
B.\ Budinsky, S.\ Smakal: Goniometricke funkce, SMM 20 [in Czech]