# Morning Session ## Problem 1 The normals to a surface all intersect a fixed straight line. Show that the surface is a portion of a surface of revolution. ## Problem 2 A uniform wire is bent into a form coinciding with the portion of the curve $y=e^x, 0 \leq x \leq a, a>1$, and the line segment $a-1 \leq x \leq a, y=$ $e^a$. The wire is then suspended from the point $\left(a-1, e^a\right)$ and a horizontal force $F$ is applied at the point $(0,1)$ to hold the wire in coincidence with the curve and segment. Assuming the $x$ axis is horizontal, show that the force $F$ is directed to the right. ## Problem 3 $A$ and $B$ are real numbers and $k$ a positive integer. Show that $\left|\frac{\cos k B \cos A-\cos k A \cos B}{\cos B-\cos A}\right|<k^2-1$whenever the left side is defined. ## Problem 4 $P(z)$ is a complex polynomial whose roots (as points in the Argand plane) can be covered by a closed circular disc of radius $R$. Show that the roots of $n P(z)-k P^{\prime}(z)$ can be covered by a closed circular disc of radius $R$ $+|k|$, where $n$ is the degree of $P(z), k$ is any complex number, and $P^{\prime}(z)$ is the derivative of $P(z)$. ## Problem 5 Given $n$ points in the plane, show that the largest distance determined by these points cannot occur more than $n$ times. ## Problem 6 $S_1=\ln a$ and $S_n=\sum_{i=1}^{n-1} \ln \left(a-S_i\right), n>1$.Show that$\lim _{n \rightarrow \infty} S_n=a-1$ ## Problem 7 Each member of a set of circles in the $x y$ plane is tangent to the $x$ axis and no two of the circles intersect. Show that: >[!i] the points of tangency can include all the rational points on the axis, but >[!ii] the points of tangency cannot include all the irrational points. # Afternoon Session ## Problem 1 Consider the determinant $\left|a_{i j}\right|$ of order 100 with $a_{i j}=i \times j$. Prove that if the absolute value of each of the 100 ! terms in the expansion of this determinant is divided by 101 then the remainder in each case is 1. ## Problem 2 If facilities for division are not available, it is sometimes convenient in determining the decimal expansion of $1 / A, A>0$ to use the iteration $X_{k+1}$ $=X_k\left(2-A X_k\right), k=0,1,2, \ldots$, where $X_0$ is a selected "starting" value. Find the limitations, if any, on the starting value $X_0$ in order that the above iteration converges to the desired value $1 / A$. ## Problem 3 For $f(x)$ a positive, monotone decreasing function defined in $0 \leq x \leq 1$ prove that$\frac{\int_0^1 x f^2(x) d x}{\int_0^1 x f(x) d x} \leq \frac{\int_0^1 f^2(x) d x}{\int_0^1 f(x) d x}$ ## Problem 4 Let $a(n)$ be the number of representations of the positive integer $n$ as the sums of 1's and 2's taking order into account. For example, since$\begin{gathered}4=1+1+2=1+2+1=2+1+1 \\=2+2=1+1+1+1\end{gathered}$then $a(4)=5$. Let $b(n)$ be the number of representations of $n$ as the sum of integers greater than 1 , again taking order into account and counting the summand $n$. For example, since $6=4+2=2+4=3+3=2+2+2$, we have $b(6)=5$. Show that for each $n, a(n)=b(n+2)$. ## Problem 5 With each subset $X$ of a set is associated a second subset $f(X)$. The association is such that whenever $X$ contains $Y$ then $f(X)$ contains $f(Y)$. Show that for some $\operatorname{set} A, f(A)=A$. ## Problem 6 The curve $y=f(x)$ passes through the origin with a slope of 1 . It satisfies the differential equation $\left(x^2+9\right) y^{\prime \prime}+\left(x^2+4\right) y=0$. Show that it crosses the $x$ axis between $ x=\frac{3}{2} \pi \text { and } x=\sqrt{\frac{63}{53}} \pi $ ## Problem 7 Let $C$ be a closed convex planar disc bounded by a regular polygon. Show that for each positive integer $n$ there exists a set of points $S(n)$ in the plane such that each $n$ points of $S(n)$ can be covered by $C$, but $S(n)$ itself cannot be covered by $C$.