We are given $b$ white balls and $n$ black balls ($b,n>0$) which are to be distributed among two urns, at least one in each. Let $s$ be the number of balls in the first urn, and $r$ the number of white ones among them. One randomly chooses an urn and randomly picks a ball from it. (a) Compute the probability $p$ that the drawn ball is white. (b) If $s$ is fixed, for which $r$ is $p$ maximal? (c) Find the distribution of balls among the urns which maximizes $p$. (d) Give a generalization for larger numbers of colors and urns.

In this problem we consider so-called cartesian triangles, that is, triangles $ABC$ with integer sides $BC=a,CA=b,AB=c$ and $\angle A=\frac{2\pi}3$. Unless noted otherwise, $\triangle ABC$ is assumed to be cartesian. (a) If $U,V,W$ are the projections of the orthocenter $H$ to $BC,CA,AB$, respectively, specify which of the segments $AU$, $BV$, $CW$, $HA$, $HB$, $HC$, $HU$, $HV$, $HW$, $AW$, $AV$, $BU$, $BW$, $CV$, $CU$ have rational length. (b) If $I$ is the incenter, $J$ the excenter across $A$, and $P,Q$ the intersection points of the two bisectors at $A$ with the line $BC$, specify those of the segments $PB$, $PC$, $QB$, $QC$, $AI$, $AJ$, $AP$, $AQ$ having rational length. (c) Assume that $b$ and $c$ are prime. Prove that exactly one of the numbers $a+b-c$ and $a-b+c$ is a multiple of $3$. (d) Assume that $\frac{a+b-c}{3c}=\frac pq$, where $p$ and $q$ are coprime, and denote by $d$ the $\gcd$ of $p(3p+2q)$ and $q(2p+q)$. Compute $a,b,c$ in terms of $p,q,d$. (e) Prove that if $q$ is not a multiple of $3$, then $d=1$. (f) Deduce a necessary and sufficient condition for a triangle to be cartesian with coprime integer sides, and by geometrical observations derive an analogous characterization of triangles $ABC$ with coprime sides $BC=a$, $CA=b$, $AB=c$ and $\angle A=\frac\pi3$.

Let $A,B,C$ be three distinct points in space, $(A)$ the sphere with center $A$ and radius $r$. Let $E$ be the set of numbers $R>0$ for which there is a sphere $(H)$ with center $H$ and radius $R$ such that $B$ and $C$ are outside the sphere, and the points of the sphere $(A)$ are strictly inside it. (a) Suppose that $B$ and $C$ are on a line with $A$ and strictly outside $(A)$. Show that $E$ is nonempty and bounded, and determine its supremum in terms of the given data. (b) Find a necessary and sufficient condition for $E$ to be nonempty and bounded (c) Given $r$, compute the smallest possible supremum of $E$, if it exists.