# Morning Session ## Problem 1 Let $n$ be an odd integer greater than 1. Let $A$ be an $n$ by $n$ symmetric matrix such that each row and each column of $A$ consists of some permutation of the integers $1, \ldots, n$. Show that each one of the integers $1, \ldots, n$ must appear in the main diagonal of $A$. ## Problem 2 Consider any five points $P_1, P_2, P_3, P_4, P_5$ in the interior of a square $S$ of side-length 1. Denote by $d_j$ the distance between the points $P_i$ and $P_j$. Prove that at least one of the distances $d_i$ is less than $\frac{\sqrt{2}}{2}$. Can $\frac{\sqrt{2}}{2}$ be replaced by a smaller number in this statement? ## Problem 3 Prove that if the family of integral curves of the differential equation $\frac{d y}{d x}+p(x) y=q(x) \quad p(x) \cdot q(x) \neq 0$ is cut by the line $x=k$, the tangents at the points of intersection are concurrent. ## Problem 4 A uniform rod of length $2 k$ and weight $w$ rests with the end $A$ against a smooth vertical wall, while to the lower end $B$ is fastened a string $B C$ of length $2 b$ coming from a point $C$ in the wall directly above $A$. If the system is in equilibrium, determine the angle $A B C$. ## Problem 5 If $f(x)$ is a real-valued function defined for $0<x<1$, then the formula $f(x)=o(x)$ is an abbreviation for the statement that $\frac{f(x)}{x} \rightarrow 0 \quad \text { as } x \rightarrow 0$ Keeping this in mind, prove the following: $\begin{array}{rl}\text{if} \qquad& \lim _{x \rightarrow 0} f(x)=0 \quad\text { and }\quad f(x)-f\left(\frac{x}{2}\right)=o(x) \\ \text {then} \qquad & f(x)=o(x)\end{array}$ ## Problem 6 Suppose that $u_0, u_1, u_2, \ldots$ is a sequence of real numbers such that $u_n=\sum_{k=1}^{\infty} u_{n+k}^2 \quad \text { for } n=0,1,2, \ldots$Prove that if $\Sigma u_n$ converges then $u_{\hbar}=0$ for all $k$. ## Problem 7 Prove that there are no integers $x$ and $y$ for which$x^2+3 x y-2 y^2=122 .$ # Afternoon Session ## Problem 1 Show that the equation $x^2-y^2=a^3$ has always integral solutions for $x$ and $y$ whenever $a$ is a positive integer. ## Problem 2 Assume as known the (true) fact that the alternating harmonic series$1-1 / 2+1 / 3-1 / 4+1 / 5-1 / 6+1 / 7-1 / 8+\cdots \tag{1}$is convergent, and denote its sum by $s$. Rearrange the series (1) as follows:$1+1 / 3-1 / 2+1 / 5+1 / 7-1 / 4+1 / 9+1 / 11-1 / 6+\cdots\tag{2}$Assume as known the (true) fact that the series (2) is also convergent, and denote its sum by $S$. Denote by $s_k, S_k$ the $k$ th partial sum of the series $(1)$ and (2) respectively. Prove the following statements. > [!i] $S_{3 n}=s_{4 n}+\frac{1}{2} s_{2 n}$, > [!ii] $S \neq s$. ## Problem 3 Let $a$ and $b$ denote real numbers such that $a<b$. The symbol $(a, b)$ will denote the closed interval with the end points $a, b$. Let there be given a collection of closed intervals $\left(a_1, b_1\right) \ldots,\left(a_n, b_n\right)$ such that any two of these closed intervals have at least one point in common. Prove that there exists then a point which is contained in every one of these intervals. ## Problem 4 Given the focus $f$ and the directrix $D$ of a parabola $P$ and a line $L$, describe (with proof) a Euclidean (i.e., ruler and compass) construction of the point or points of intersection of $L$ and $P$. Be sure to identify the case for which there are no points of intersection. ## Problem 5 Let $f(x)$ be a real-valued function, defined for $-1<x<1$, such that $f^{\prime}(0)$ exists. Let $a_n, b_n$ be two sequences such that$-1<a_n<0<b_n<1, \lim _{n \rightarrow \infty} a_n=0, \lim _{n \rightarrow \infty} b_n=0$. Prove that $\lim _{n \rightarrow \infty} \frac{f\left(b_n\right)-f\left(a_n\right)}{b_n-a_n}=f^{\prime}(0).$ ## Problem 6 Prove that every positive rational number is the sum of a finite number of distinct terms of the series$1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}+\cdots .$ ## Problem 7 Show that$\lim _{n \rightarrow \infty} \sum_{s=1}^n\left(\frac{a+s}{n}\right)^n \quad(a>0)$