Putnam 1949

Answer either (i) or (ii):

(i) Let $a>0.$ Three straight lines pass through the three points $(0,-a,a), (a,0,-a)$ and $(-a,a,0),$ parallel to the $x-,y-$ and $z-$ axis, respectively. A variable straight line moves so that it has one point in common with each of the three given lines. Find the equation of the surface described by the variable line.

(II) Which planes cut the surface $xy+yz+xz=0$ in (1) circles, (2) parabolas?

We consider three vectors drawn from the same initial point $O,$ of lengths $a,b$ and $c$ , respectively. Let $E$ be the parallelepiped with vertex $O$ of which the given vectors are the edges and $H$ the parallelepiped with vertex $O$ of which the given vectors are the altitudes. Show that the product of the volumes of $E$ and $H$ equals $(abc)^{2}$ and generalize this result to $n$ dimensions.

Assume that the complex numbers $a_1 , a_2, \ldots$ are all different from $0$ , and that $|a_r - a_s| >1$ for $r\ne s.$ Show that the series $$\sum_{n=1}^{\infty} \frac{1}{a_{n}^{3}}$$ converges.

Given that $P$ is a point inside a tetrahedron with vertices at $A, B, C$ and $D$ , such that the sum of the distances $PA+PB+PC+PD$ is a minimum, show that the two angles $\angle APB$ and $\angle CPD$ are equal and are bisected by the same straight line. What other pair of angles must be equal?

How many roots of the equation $z^6 +6z +10=0$ lie in each quadrant of the complex plane?

Prove that for every real or complex $x$ $$\prod_{k=1}^{\infty} \frac{1+2\cos \frac{2x}{3^{k}}}{3} =\frac{\sin x}{x}.$$

Each rational number $\frac{p}{q}$ with $p,q$ coprime of the open interval $(0,1)$ is covered by the closed interval $\left[\frac{p}{q}-\frac{1}{4q^{2}}, \frac{p}{q}+\frac{1}{4q^{2}}\right]$ . Prove that $\frac{\sqrt{2}}{2}$ is not covered by any of the above closed intervals.

Let $K$ be a closed plane curve such that the distance between any two points of $K$ is always less than $1.$ Show that $K$ lies in a circle of radius $\frac{1}{\sqrt{3}}.$

Show that the coefficients $a_1 , a_2 , a_3 ,\ldots$ in the expansion $$\frac{1}{4}\left(1+x-\frac{1}{\sqrt{1-6x+x^{2}}}\right) =a_{1} x+ a_2 x^2 + a_3 x^3 +\ldots$$ are positive integers.

let $(a_{n})$ be an arbitrary sequence of positive numbers. Show that $$\limsup_{n\to \infty} \left(\frac{a_1 +a_{n+1}}{a_{n}}\right)^{n} \geq e.$$

Let $C$ be a closed convex curve with a continuously turning tangent and let $O$ be a point inside $C.$ For each point $P$ on $C$ we define $T(P)$ as follows: Draw the tangent to $C$ at $P$ and from $O$ drop the perpendicular to that tangent. Then $T(P)$ is the point at which $C$ intersects this perpendicular. Starting now with a point $P_{0}$ on $C$ , define points $P_n$ by $P_n =T(P_{n-1}).$ Prove that the points $P_{n}$ approach a limit and characterize all possible limit points. (You may assume that $T$ is continuous.)