Each vertex of a regular $1997$-gon is labeled with an integer, so that the sum of the integers is $1$. We write down the sums of the first $k$ integers read counterclockwise, starting from some vertex $(k=1,2,\ldots,1997)$. Can we always choose the starting vertex so that all these sums are positive? If yes, how many possible choices are there?

A region in space is determined by a sphere with center $O$ and radius $R$, and a cone with vertex $O$ which intersects the sphere in a circle of radius $r$. Find the maximum volume of a cylinder contained in this region, having the same axis as the cone.

Let $C$ be a unit cube and let $p$ denote the orthogonal projection onto the plane. Find the maximum area of $p(C)$.

In a triangle $ABC$, let $a,b,c$ be its sides and $m,n,p$ be the corresponding medians. For every $\alpha>0$, let $\lambda(\alpha)$ be the real number such that $$a^\alpha+b^\alpha+c^\alpha=\lambda(\alpha)^\alpha\left(m^\alpha+n^\alpha+p^\alpha\right)^\alpha.$$(a) Compute $\lambda(2)$. (b) Find the limit of $\lambda(\alpha)$ as $\alpha$ approaches $0$. (c) For which triangles $ABC$ is $\lambda(\alpha)$ independent of $\alpha$?

Given two distinct points $A,B$ in the plane, for each point $C$ not on the line $AB$, we denote by $G$ and $I$ the centroid and incenter of the triangle $ABC$, respectively. (a) For $0<\alpha<\pi$, let $\Gamma$ be the set of points $C$ in the plane such that $\angle\left(\overrightarrow{CA},\overrightarrow{CB}\right)=\alpha+2k\pi$ as an oriented angle, where $k\in\mathbb Z$. If $C$ describes $\Gamma$, show that points $G$ and $I$ also descibre arcs of circles, and determine these circles. (b) Suppose that in addition $\frac\pi3<\alpha<\pi$. For which positions of $C$ in $\Gamma$ is $GI$ minimal? (c) Let $f(\alpha)$ denote the minimal $GI$ from the part (b). Give $f(\alpha)$ explicitly in terms of $a=AB$ and $\alpha$. Find the minimum value of $f(\alpha)$ for $\alpha\in\left(\frac\pi3,\pi\right)$.