

1. Solve the equations:
x_{1} + 2 x_{2} + 3 x_{3} + ... + (n1) x_{n1} + n x_{n} = n
2 x_{1} + 3 x_{2} + 4 x_{3} + ... + n x_{n1} + x_{n} = n1
3 x_{1} + 4 x_{2} + 5 x_{3} + ... + x_{n1} + 2 x_{n} = n2
...
(n1) x_{1} + n x_{2} + x_{3} + ... + (n3) x_{n1} + (n2) x_{n} = 2
n x_{1} + x_{2} + 2 x_{3} + ... + (n2) x_{n1} + (n1) x_{n} = 1.


2. Find rational x in (3, 4) such that √(x3) and √(x+1) are rational.


3. Express x^{13} + 1/x^{13} as a polynomial in y = x + 1/x.


4. f(x) is continuous on the interval [0, π] and satisfies ∫ _{0}^{π} f(x) dx = 0, ∫ _{0}^{π} f(x) cos x dx = 0. Show that f(x) has at least two zeros in the interval (0, π).


5. Find the smallest positive integer a such that for some integers b, c the polynomial ax^{2}  bx + c has two distinct zeros in the interval (0, 1).


6. Find the sharpest inequalities of the form a·AB < AG < b·AB and c·AB < BG < d·AB for all triangles ABC with centroid G such that GA > GB > GC.

