VMO 1965

At a time $t = 0$ , a navy ship is at a point $O$ , while an enemy ship is at a point $A$ cruising with speed $v$ perpendicular to $OA = a$ . The speed and direction of the enemy ship do not change. The strategy of the navy ship is to travel with constant speed $u$ at a angle $0 < \phi < \pi /2$ to the line $OA$ .

1. Let $\phi$ be chosen. What is the minimum distance between the two ships? Under what conditions will the distance vanish?
2. If the distance does not vanish, what is the choice of $\phi$ to minimize the distance? What are directions of the two ships when their distance is minimum?

$AB$ and $CD$ are two fixed parallel chords of the circle $S$ . $M$ is a variable point on the circle. $Q$ is the intersection of the lines $MD$ and $AB$ . $X$ is the circumcenter of the triangle $MCQ$ . Find the locus of $X$ . What happens to $X$ as $M$ tends to (1) $D$ , (2) $C$ ? Find a point $E$ outside the plane of $S$ such that the circumcenter of the tetrahedron $MCQE$ has the same locus as $X$ .

1. Two nonnegative real numbers $x, y$ have constant sum $a$ . Find the minimum value of $x^m + y^m$ , where m is a given positive integer.
2. Let $m, n$ be positive integers and $k$ a positive real number. Consider nonnegative real numbers $x_1, x_2, . . . , x_n$ having constant sum $k$ . Prove that the minimum value of the quantity $x^m_1+ ... + x^m_n$ occurs when $x_1 = x_2 = ... = x_n$ .