ÖMO 1970

Let $x,y,z$ be positive real numbers such that $x+y+z=1$ Prove that always $\left( 1+\frac1x\right)\times\left(1+\frac1y\right)\times\left(1 +\frac1z\right)\ge 64$ When does equality hold?

In the plane seven different points $P_1, P_2, P_3, P_4, Q_1, Q_2, Q_3$ are given. The points $P_1, P_2, P_3, P_4$ are on the straight line $p$ , the points $Q_1, Q_2, Q_3$ are not on $p$ . By each of the three points $Q_1, Q_2, Q_3$ the perpendiculars are drawn on the straight lines connecting points different of them. Prove that the maximum's number of the possibles intersections of all perpendiculars is to 286, if the points $Q_1, Q_2, Q_3$ are taken in account as intersections.

$E_1$ and $E_2$ are parallel planes and their distance is $p$ . (a) How long is the seitenkante of the regular octahedron such that a side lies in $E_1$ and another in $E_2$ ? (b) $E$ is a plane between $E_1$ and $E_2$ , parallel to $E_1$ and $E_2$ , so that its distances from $E_1$ and $E_2$ are in ratio $1:2$ Draw the intersection figure of $E$ and the octahedron for $P=4\sqrt{\frac32}$ cm and justifies, why the that figure must look in such a way

Find all real solutions of the following set of equations: $$72x^3+4xy^2=11y^3$$ $$27x^5-45x^4y-10x^2y^3=\frac{-143}{32}y^5$$