# Morning Session ## Problem 1 Given five points in a plane, no three of which lie on a straight line, show that some four of these points form the vertices of a convex quadrilateral. ## Problem 2 Find every real-valued function $f$ whose domain is an interval $I$ (finite or infinite) having 0 as a left-hand endpoint, such that for every positive member $x$ of $I$ the average of $f$ over the closed interval $[0, x]$ is equal to the geometric mean of the numbers $f(0)$ and $f(x)$. ## Problem 3 In a triangle $A B C$ in the Euclidean plane, let $A^{\prime}$ be a point on the segment from $B$ to $C, B^{\prime}$ a point on the segment from $C$ to $A$, and $C^{\prime}$ a point on the segment from $A$ to $B$ such that $\frac{A B^{\prime}}{B^{\prime} C}=\frac{B C^{\prime}}{C^{\prime} A}=\frac{C A^{\prime}}{A^{\prime} B}=k,$ where $k$ is a positive constant. Let $\Delta$ be the triangle formed by parts of the segments obtained by joining $A$ and $A^{\prime}, B$ and $B^{\prime}$, and $C$ and $C^{\prime}$. Prove that the areas of the triangles $\Delta$ and $A B C$ are in the ratio. $\frac{(k-1)^2}{k^2+k+1}$ ## Problem 4 Assume that $|f(x)| \leq 1$ and $\left|f^{\prime \prime}(x)\right| \leq 1$ for all $x$ on an interval of length at least 2. Show that $\left|f^{\prime}(x)\right| \leq 2$ on the interval. ## Problem 5 Evaluate in closed form $\sum_{k=1}^n\binom{n}{k} k^2 .$ Note $\binom{n}{k}=\frac{n(n-1) \cdots(n-k+1)}{1 \cdot 2 \cdots k} .$ ## Problem 6 Let $S$ be a set of rational numbers such that whenever $a$ and $b$ are members of $S$, so are $a+b$ and $a b$, and having the property that for every rational number $r$ exactly one of the following three statements is true: $r \in S,-r \in S, \quad r=0.$ Prove that $S$ is the set of all positive rational numbers. # Afternoon Session ## Problem 1 Let $x^{(n)}=x(x-1) \cdots(x-n+1)$ for $n$ a positive integer and let $x^{(0)}$ $=1$. Prove that $(x+y)^{(n)}=\sum_{k=0}^n\binom{n}{k} x^{(k)} y^{(n-k)}.$ Note $\binom{n}{k}=\frac{n(n-1) \cdots(n-k+1)}{1 \cdot 2 \cdots k} .$ ## Problem 2 Let $R$ be the set of all real numbers and $S$ the set of all subsets of the positive integers. Construct a function $f$ whose domain is $R$ and whose range is in $S$, such that $f(a)$ is a proper subset of $f(b)$ whenever $a<b$. ## Problem 3 Let $S$ be a convex region in the Euclidean plane containing the origin. Assume that every ray (that is, half-line) from the origin has at least one point outside $S$. Prove that $S$ is bounded. (A region in the plane is defined to be convex if and only if the line segment joining every pair of its points lies entirely within the region.) ## Problem 4 The Euclidean plane is divided into regions by drawing a finite number of circles. Show that it is possible to color each of these regions either red or blue in such a way that no two adjacent regions have the same color. (Two such regions are said to be adjacent if and only if their boundaries have an arc of a circle in common.) ## Problem 5 Prove that for every integer $n$ greater than $1$: $\frac{3 n+1}{2 n+2}<\left(\frac{1}{n}\right)^n+\left(\frac{2}{n}\right)^n+\cdots+\left(\frac{n}{n}\right)^n<2$ ## Problem 6 Let $f(x)=\sum_{k=0}^n a_k \sin k x+b_k \cos k x,$ where $a_k$ and $b_k$ are constants. Show that, if $|f(x)| \leq 1$ for $0 \leq x \leq 2 \pi$ and $\left|f\left(x_i\right)\right|=1$ for $0 \leq x_1<x_2<\cdots<x_{2 n}<2 \pi$, then $f(x)=\cos (n x+a)$ for some constant $a$.