SE-SKOLORNAS 1979

Solve the equations: $$\left\{ \begin{array}{l} x_1 + 2 x_2 + 3 x_3 + \cdots + (n-1) x_{n-1} + n x_n = n 2 x_1 + 3 x_2 + 4 x_3 + \cdots + n x_{n-1} + x_n = n-1 3 x_1 + 4 x_2 + 5 x_3 + \cdots + x_{n-1} + 2 x_n = n-2 \cdots \cdots \cdots \cdots\cdot\cdots \cdots \cdots \cdots\cdot\cdots \cdots \cdots \cdots\cdot (n-1) x_1 + n x_2 + x_3 + \cdots + (n-3) x_{n-1} + (n-2) x_n = 2 n x_1 + x_2 + 2 x_3 + \cdots + (n-2) x_{n-1} + (n-1) x_n = 1 \end{array} \right.$$

Find rational $x$ in $(3,4)$ such that $\sqrt{x-3}$ and $\sqrt{x+1}$ are rational.

Express $$x^{13} + \frac{1}{x^{13}}$$ as a polynomial in $y = x + \frac{1}{x}$ .

$f(x)$ is continuous on the interval $[0, \pi]$ and satisfies $$\int\limits_0^\pi f(x)dx=0, \int\limits_0^\pi f(x)\cos x dx=0$$ Show that $f(x)$ has at least two zeros in the interval $(0, \pi)$ .

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 $(0,1)$ .

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