Putnam 1948

What is the maximum of $|z^3 -z+2|$ , where $z$ is a complex number with $|z|=1?$

Two spheres in contact have a common tangent cone. These three surfaces divide the space into various parts, only one of which is bounded by all three surfaces, it is "ring-shaped." Being given the radii of the spheres, $r$ and $R$ , find the volume of the "ring-shaped" part. (The desired expression is a rational function of $r$ and $R.$ )

Let $(a_n)$ be a decreasing sequence of positive numbers with limit $0$ such that $$b_n = a_n -2 a_{n+1}+a_{n+2} \geq 0$$ for all $n.$ Prove that $$\sum_{n=1}^{\infty} n b_n =a_1.$$

Let $D$ be a plane region bounded by a circle of radius $r.$ Let $(x,y)$ be a point of $D$ and consider a circle of radius $\delta$ and center at $(x,y).$ Denote by $l(x,y)$ the length of that arc of the circle which is outside $D.$ Find $$\lim_{\delta \to 0} \frac{1}{\delta^{2}} \int_{D} l(x,y) dx dy.$$

If $\xi_1,\ldots,\xi_n$ denote the $n$ -th roots of unity, evaluate $$\prod_{1\leq i<j \leq n} (\xi_{i}-\xi_j )^2 .$$

Answer either (i) or (ii):

(i) A force acts on the element $ds$ of a closed plane curve. The magnitude of this force is $r^{-1} ds$ where $r$ is the radius of curvature at the point considered, and the direction of the force is perpendicular to the curve, it points to the convex side. Show that the system of such forces acting on all elements of the curve keep it in equilibrium.

(ii) Show that $$x+ \frac{2}{3}x^{3}+ \frac{2\cdot 4}{3\cdot 5} x^5 +\frac{2\cdot 4\cdot 6}{3\cdot 5\cdot 7} x^7 + \ldots= \frac{ \arcsin x}{\sqrt{1-x^{2}}}.$$

Let $f(x)$ be a cubic polynomial with roots $x_1 , x_2$ and $x_3.$ Assume that $f(2x)$ is divisible by $f'(x)$ and compute the ratio $x_1 : x_2: x_3 .$

A circle moves so that it is continually in the contact with all three coordinate planes of an ordinary rectangular system. Find the locus of the center of the circle.

For what $\lambda$ does the equation $$\int_{0}^{1} \min(x,y) f(y) dy =\lambda f(x)$$ have continuous solutions which do not vanish identically in $(0,1)?$ What are these solutions?

The pairs $(a,b)$ such that $|a+bt+ t^2 |\leq 1$ for $0\leq t \leq 1$ fill a certain region in the plane. What is the area of this region?

Answer wither (i) or (ii):

(i) Let $V, V_1 , V_2$ and $V_3$ denote four vertices of a cube such that $V_1 , V_2 , V_3$ are adjacent to $V.$ Project the cube orthogonally on a plane of which the points are marked with complex numbers. Let the projection of $V$ fall in the origin and the projections of $V_1 , V_2 , V_3$ in points marked with the complex numbers $z_1 , z_2 , z_3$ , respectively. Show that $z_{1}^{2} +z_{2}^{2} +z_{3}^{2}=0.$ (ii) Let $(a_{ij})$ be a matrix such that $$|a_{ii}| > |a_{i1}| + |a_{i2}|+\ldots +|a_{i i-1}|+ |a_{i i+1}| +\ldots +|a_{in}|$$ for all $i.$ Show that the determinant is not equal to $0.$