# Morning Session ## Problem 1 Given a set of 6 points in the plane, prove that the ratio of the longest distance between any pair to the shortest is at least $\sqrt{3}$. ## Problem 2 Find all continuous positive functions $f(x)$, for $0 \leq x \leq 1$, such that $ \begin{gathered} \int_0^1 f(x) d x=1 \\ \int_0^1 f(x) x d x=\alpha \\ \int_0^1 f(x) x^2 d x=\alpha^2 \end{gathered} $ where $\alpha$ is a given real number. ## Problem 3 Let $P_1, P_2, \ldots$ be a sequence of distinct points which is dense in the interval $(0,1)$. The points $P_1, P_2, \ldots, P_{n-1}$ decompose the interval into $n$ parts, and $P_n$ decomposes one of these into two parts. Let $a_n$ and $b_n$ be the lengths of these two intervals. Prove that $ \sum_{n=1}^{\infty} a_n b_n\left(a_n+b_n\right)=1 / 3 . $ (A sequence of points in an interval is said to be dense when every subinterval contains at least one point of the sequence.) ## Problem 4 Let $p_n(n=1,2, \ldots)$ be a bounded sequence of integers which satisfies the recursion $ p_n=\frac{p_{n-1}+p_{n-2}+p_{n-3} p_{n-4}}{p_{n-1} p_{n-2}+p_{n-3}+p_{n-4}} . $ Show that the sequence eventually becomes periodic. ## Problem 5 Prove that there is a constant $K$ such that the following inequality holds for any sequence of positive numbers $a_1, a_2, a_3, \ldots$ : $ \sum_{n=1}^{\infty} \frac{n}{a_1+a_2+\cdots+a_n} \leq K \sum_{n=1}^{\infty} \frac{1}{a_n} . $ ## Problem 6 Let $S$ be a finite subset of a straight line. Say that $S$ has the repeated distance property when every value of the distance between pairs of points of $S$ (except for the longest) occurs at least twice. Show that if $S$ has the repeated distance property then the ratio of any two distances between two points of $S$ is a rational number. # Afternoon Session ## Problem 1 Let $u_k(k=1,2, \ldots)$ be a sequence of integers, and let $V_n$ be the number of those which are less than or equal to $n$. Show that if $\sum_{k=1}^{\infty} 1 / u_k<\infty,$then$\lim _{n \rightarrow \infty} V_n / n=0 .$ ## Problem 2 Let $S$ be a set of $n>0$ elements, and let $A_1, A_2, \ldots, A_k$ be a family of distinct subsets, with the property that any two of these subsets meet. Assume that no other subset of $S$ meets all of the $A_i$. Prove that $k=2^{n-1}$. ## Problem 3 Let $f(x)$ be a real continuous function defined for all real $x.$ Assume that for every $\epsilon>0$ $\lim _{n \rightarrow \infty} f(n \epsilon)=0,\quad \text{(where $n$ is a positive integer).}$ Prove that$\lim _{x \rightarrow \infty} f(x)=0 .$ ## Problem 4 Into how many regions do $n$ great circles (no three concurrent) decompose the surface of the sphere on which they lie? ## Problem 5 Let $u_n\ (n=1,2,3, \ldots)$ denote the least common multiple of the first $n$ terms of a strictly increasing sequence of positive integers (for example, the sequence $1,2,3,4,5,6,10,12, \ldots)$. Prove that the series $\sum_{n=1}^{\infty} 1 / u_n$ is convergent. ## Problem 6 Show that the unit disk in the plane cannot be partitioned into two disjoint congruent subsets.