Putnam 1957

The normals to a surface all intersect a fixed straight line. Show that the surface is a portion of a surface of revolution.

Let $a>1.$ A uniform wire is bent into a form coinciding with the portion of the curve $y=e^x$ for $x\in [0,a]$ , and the line segment $y=e^a$ for $x\in [a-1,a].$ The wire is then suspended from the point $(a-1, e^a)$ and a horizontal force $F$ is applied to the point $(0,1)$ to hold the wire in coincidence with the curve and segment. Show that the force $F$ is directed to the right.

Let $a,b$ be real numbers and $k$ a positive integer. Show that $$\left| \frac{ \cos kb \cos a - \cos ka \cos b}{\cos b -\cos a} \right|<k^2 -1$$ whenever the left side is defined.

Given $n$ points in the plane, show that the largest distance determined by these points cannot occur more than $n$ times.

Let $a>0$ , $S_1 =\ln a$ and $S_n = \sum_{i=1 }^{n-1} \ln( a- S_i )$ for $n >1.$ Show that $$\lim_{n \to \infty} S_n = a-1.$$

Consider the determinant of the matrix $(a_{ij})_{ij}$ with $1\leq i,j \leq 100$ and $a_{ij}=ij.$ 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 is always $1.$

In order to determine $\frac{1}{A}$ for $A>0$ , one can use the iteration $X_{k+1}=X_{k}(2-AX_{k}),$ where $X_0$ is a selected starting value. Find the limitation, if any, on the starting value $X_0$ so that the above iteration converges to $\frac{1}{A}.$

For $f(x)$ a positive , monotone decreasing function defined in $[0,1],$ prove that $$\int_{0}^{1} f(x) dx \cdot \int_{0}^{1} xf(x)^{2} dx \leq \int_{0}^{1} f(x)^{2} dx \cdot \int_{0}^{1} xf(x) dx.$$

Let $a(n)$ be the number of representations of the positive integer $n$ as an ordered sum of $1$ 's and $2$ 's. Let $b(n)$ be the number of representations of the positive integer $n$ as an ordered sum of integers greater than $1.$ Show that $a(n)=b(n+2)$ for each $n$ .

The curve $y=y(x)$ satisfies $y'(0)=1.$ It satisfies the differential equation $(x^2 +9)y'' +(x^2 +4)y=0.$ Show that it crosses the $x$ -axis between $$x= \frac{3}{2} \pi \;\; \text{and} \;\; x= \sqrt{\frac{63}{53}} \pi.$$