# Morning Session ## Problem 1 Let $f(m, 1)=f(1, n)=1$ for $m \geq 1, n \geq 1$, and let $f(m, n)=$ $f(m-1, n)+f(m, n-1)+f(m-1, n-1)$ for $m>1$ and $n>1$. Also let $S(n)=\sum_{e+b=n} f(a, b), \quad a \geq 1 \text { and } b \geq 1 .$ Prove that $S(n+2)=S(n)+2 S(n+1) \quad \text { for } n \geq 2 .$ ## Problem 2 Let $R_1=1, \quad R_{n+1}=1+n / R_n, \quad n \geq 1 .$ Show that for $n \geq 1$, $\sqrt{n} \leq R_n \leq \sqrt{n}+1.$ ## Problem 3 Under the assumption that the following set of relations has a unique solution for $u(t),$ determine it. $ \begin{aligned} & \frac{d u(t)}{d t}=u(t)+\int_0^4 u(s) d s, \\ & u(0)=1 . \end{aligned} $ ## Problem 4 In assigning dormitory rooms, a college gives preference to pairs of $A A, A B, A C, B B, B C, A D, C C, B D, C D, D D,$ in which $A A$ means two seniors, $A B$ means a senior and a junior, etc. Determine numerical values to assign to $A, B, C, D$ so that the set of numbers $A+A, A+B, A+C, B+B$, etc., corresponding to the order above will be in descending magnitude. Find the general solution and the solution in least positive integers. ## Problem 5 Show that the number of non-zero terms in the expansion of the $n$th order determinant having zeros in the main diagonal and ones elsewhere is $n\left[1\left[-\frac{1}{11}+\frac{1}{21}-\frac{1}{31}+\cdots+\frac{(-1)^2}{n !}\right] \cdot(\right.$ (pese (sin) 6. Let $a(x)$ and $b(x)$ be continuous functions on $0 \leq x \leq 1$ and let $0 \leq$ $a(x) \leq a<1$ on that range. Under what other conditions (if any) is the solu$a(x) \leq a<1$ on that range. tion of the equation for $u$, $u=\operatorname{maximum}_{0 \leq x \leq 1}[b(x)+a(x) \cdot u]$, given by $ u=\underset{0 x+1}{\operatorname{maximum}}\left[\frac{b(x)}{1-a(x)}\right] ? $ ## Problem 7 Let $a$ and $b$ be relatively prime positive integers, $b$ even. For each positive integer $q$ let $p=p(q)$ be chosen so that $ \left|\frac{p}{q}-\frac{a}{b}\right| $ is a minimum. Prove that $ \lim _{n \rightarrow \infty} \sum_{q=1}^n \frac{q\left|\frac{p}{q}-\frac{a}{b}\right|}{n}=\frac{1}{4} . $ # Afternoon Session ## Problem 1 Given $ b_n=\sum_{k=0}^n\left(\begin{array}{l} n \\ k \end{array}\right)^{-1}, \quad n \geq 1 $ prove that $ b_n=\frac{n+1}{2 n} b_{n-1}+1, \quad n \geq 2 $ and hence, as a corollary, $ \lim _{n \rightarrow \infty} b_n=2 $ ## Problem 2 Given a set of $n+1$ positive integers, none of which exceeds $2 n$, show that at least one member of the set must divide another member of the set. ## Problem 3 If a square of unit side be partitioned into two sets, then the diameter (least upper bound of the distances between pairs of points) of one of the sets is not less than $\sqrt{5} / 2$. Show also that no larger number will do. ## Problem 4 Let $C$ be a real number, and let $f$ be a function such that $\lim _{x \rightarrow \infty} f(x)=C, \qquad\qquad \lim _{x \rightarrow \infty} f^{\prime \prime}(x)=0.$ Prove that $\lim _{x \rightarrow \infty} f^{\prime}(x)=0\text{\quad and\quad}\lim _{x \rightarrow \infty} f^{\prime \prime}(x)=0,$ where superscripts denote derivatives. ## Problem 5 The lengths of successive segments of a broken line are represented by the successive terms of the harmonic progression $1,1 / 2,1 / 3, \ldots, 1 / n, \ldots.$ Each segment makes with the preceding segment a given angle $\theta$. What is the the initial point of the first segment? ## Problem 6 Let a complete oriented graph on $n$ points be given, i.e., a set of $n$ points $1,2,3, \ldots, n$, and between any two points $i$ and $j$ a direction, $i-j$. Show that there exists a permutation of the points, $\left[a_1, a_2, a_3, \ldots, a_n\right]$, such that $a_1-a_2-a_3-\cdots-a_n$. ## Problem 7 Let $a_1, a_2, \ldots, a_n$ be a permutation of the integers $1,2, \ldots, n$, Call $a$ a "big" integer if $a_i>a_j$ for all $j>i$. Find the mean number of "big" integers over all permutations on the first $n$ positive integers.