VMO 1975

The roots of the equation $x^3 - x + 1 = 0$ are $a, b, c$ . Find $a^8 + b^8 + c^8$ .

Find all real x which satisfy ($x^3$ + $a^3$)/(x + a)$^3$ + ($x^3$ + $b^3$)/(x + b)$^3$ + ($x^3$ + $c^3$)/(x + c)$^3$ + 3(x - a)(x - b)(x - c)/( 2(x + a)(x + b)(x + c) ) = 3/2.

Let $ABCD$ be a tetrahedron with $BA \perp AC,DB \perp (BAC)$ . Denote by $O$ the midpoint of $AB$ , and $K$ the foot of the perpendicular from $O$ to $DC$ . Suppose that $AC = BD$ . Prove that $\frac{V_{KOAC}}{V_{KOBD}}=\frac{AC}{BD}$ if and only if $2AC \cdot BD = AB^2$ .

Find all terms of the arithmetic progression $-1, 18, 37, 56, ...$ whose only digit is $5$ .

Show that the sum of the (local) maximum and minimum values of the function $\frac{tan(3x)}{tan^3x}$ on the interval $\big(0, \frac{\pi }{2}\big)$ is rational.

L is a fixed line and A a fixed point not on L. L' is a variable line (in space) through A. Let M be the point on L and N the point on L' such that MN is perpendicular to L and L'. Find the locus of M and the locus of the midpoint of MN.