# Morning Session ## Problem 1 What is the maximum of $\left|z^3-z+2\right|$; where $z$ is a complex number with $|z|=1$ ? > [!sol]- Click for Solution! > Let $f(z)=z^3-z+2$. We may as well maximize $|f(z)|^2$. If $|z|=1$, then $z=x+i y$, where $y^2=1-x^2$ and $-1 \leq x \leq 1$, so >$\begin{aligned}|f(z)|^2 & =\left|(x+i y)^3-(x+i y)+2\right|^2 \\& =\left|x^3-3 x\left(1-x^2\right)-x+2+i y\left(3 x^2-\left(1-x^2\right)-1\right)\right|^2 \\& =\left(4 x^3-4 x+2\right)^2+\left(1-x^2\right)\left(4 x^2-2\right)^2 \\& =16 x^3-4 x^2-16 x+8=L(x)\end{aligned}$ > Hence we seek $\max _{-1 \leq x \leq 1} L(x)$. This maximum must be attained either at a critical point or at an endpoint. The critical points, obtained by solving $L^{\prime}(x)=48 x^2-8 x-16=0$, are $x=-\frac{1}{2}, \frac{2}{3}$. Since > $L(-1)=4, \quad L\left(-\frac{1}{2}\right)=13, \quad L\left(\frac{2}{3}\right)=\frac{8}{27}, \quad L(1)=4$ > the maximum value of $L$ is 13 , attained for $x=-\frac{1}{2}$. Hence the maximum value of $|f(z)|$ on the unit circle is $\sqrt{13}$, attained when $\operatorname{Re} z=-\frac{1}{2}$, i.e., when $z=(-1 \pm i \sqrt{3}) / 2$ ## Problem 2 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 $\boldsymbol{R}$, find the volume of the "ring-shaped" part. (The desired expression is a rational function of $r$ and $R$.) ## Problem 3 Let $\left\{a_n\right\}$ 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 . $ ## Problem 4 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 _{d \rightarrow 0} \frac{1}{\delta^2} \iint_D l(x, y) d x d y $ Note. This is the text of the problem as published in the American Mathematical Monthly, vol. 55 (1948) p. 632. Clearly the " $d$ " under the limit sign should be a " $\delta . "$ " We do not know how the problem was printed on the actual examination. ## Problem 5 If $x_1, \ldots, x_n$ denote the $n$th roots of unity, evaluate $ \begin{array}{lll} \pi\left(x_i-x_j\right)^2 & (i<j) . \end{array} $