Question bank VMO 1965

At the initial time $t=0$, our naval ship at $O$ detects an enemy ship at $A$, where $OA=a$, moving with constant speed $v$ along a straight line perpendicular to $OA$. To pursue and fire on it, our ship starts moving with constant speed $u$ along a straight line making an acute angle $\varphi$ with $OA$.

(a) For a preselected $\varphi$, what is the minimum distance between the ships? What condition on $u,v$ makes this minimum zero?

(b) If that condition is not satisfied, how should $\varphi$ be chosen to make this minimum distance as small as possible? At closest approach, what angle should our firing direction make with the enemy's path?

A circle has two parallel chords $AB$ and $CD$. A point $M$ moves on the circle, and the line $MD$ meets the line $AB$ at $Q$.

(a) As $M$ tends to $D$ or to $C$, where does the circumcenter of triangle $MCQ$ tend? Determine the locus of the center of $(MCQ)$.

(b) Take a fixed point $E$ outside the plane of the diagram. How should $E$ be chosen so that the locus of the center of the sphere $(MCQE)$ coincides with the locus of the center of $(MCQ)$?

(a) Two nonnegative numbers $x,y$ have fixed sum $a$. Determine $x,y$ for which $x^m+y^m$ is smallest, where $m$ is a given natural number. Given $n$ nonnegative numbers with fixed sum, prove that for $x_1^m+x_2^m+\cdots+x_n^m$ to be smallest one must have $x_1=x_2=\cdots=x_n$.

(b) One cubic meter of a chemical powder is to be stored in eight cubic wooden boxes. Every flat wooden panel has the same fixed, negligible thickness. The price of each square panel of that thickness is proportional to the square of its area. How should the edges $x_1,x_2,\ldots,x_8$ be chosen to minimize the cost of wood?