VMO 1979

Show that for all $x > 1$ there is a triangle with sides, $x^4 + x^3 + 2x^2 + x + 1, 2x^3 + x^2 + 2x + 1, x^4 - 1.$

Find all real numbers $a, b, c$ such that $x^3 + ax^2 + bx + c$ has three real roots $\alpha, \beta,\gamma$ (not necessarily all distinct) and the equation $x^3 + \alpha^3 x^2 + \beta^3 x + \gamma^3$ has roots $\alpha^3, \beta^3,\gamma^3$ .

$ABC$ is a triangle. Find a point $X$ on $BC$ such that : area $ABX$ / area $ACX$ = perimeter $ABX$ / perimeter $ACX$ .

For each integer $n > 0$ show that there is a polynomial $p(x)$ such that $p(2 cos x) = 2 cos nx$ .

Find all real numbers $k$ such that $x^2 - 2 x [x] + x - k = 0$ has at least two non-negative roots.

$ABCD$ is a rectangle with $BC / AB = \sqrt2$ . $ABEF$ is a congruent rectangle in a different plane. Find the angle $DAF$ such that the lines $CA$ and $BF$ are perpendicular. In this configuration, find two points on the line $CA$ and two points on the line $BF$ so that the four points form a regular tetrahedron.