Putnam 1954

Let $n$ be an odd integer greater than $1.$ Let $A$ be an $n\times n$ symmetric matrix such that each row and column consists of some permutation of the integers $1,2, \ldots, n.$ Show that each of the integers $1,2, \ldots, n$ must appear in the main diagonal of $A$ .

Prove that if the family of integral curves of the differential equation $$\frac{dy}{dx} +p(x) y= q(x),$$ where $p(x) q(x) \ne 0$ , is cut by the line $x=k$ the tangents at the points of intersection are concurrent.

A uniform rod of length $2k$ and weight $w$ rests with the end $A$ against a vertical wall, while the lower end $B$ is fastened by a string $BC$ of length $2b$ coming from a point $C$ in the wall above $A.$ If the system is in equilibrium, determine the angle $ABC.$

Let $f(x)$ be a real-valued function defined for $0<x<1.$ If $$\lim_{x \to 0} f(x) =0 \; \text{and} \; f(x) - f \left( \frac{x}{2} \right) =o(x),$$ prove that $f(x) =o(x),$ where we use the O-notation.

Suppose that $u_0 , u_1 ,\ldots$ is a sequence of real numbers such that $$u_n = \sum_{k=1}^{\infty} u_{n+k}^{2}\;\; \text{for} n=0,1,2,\ldots$$ Prove that if $\sum u_n$ converges, then $u_k=0$ for all $k$ .

Show that the equation $x^2 -y^2 =a^3$ has always integral solutions for $x$ and $y$ whenever $a$ is a positive integer.

Let $s$ denote the sum of the alternating harmonic series. Rearrange this series as follows $$1 + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} +\frac{1}{7} - \frac{1}{4} + \frac{1}{9} + \frac{1}{11} - \ldots$$ Assume as known that this series converges as well and denote its sum by $S$ . Denote by $s_k, S_k$ respectively the $k$ -th partial sums of both series. Prove that $$\!\!\! \text{i}) S_{3n} = s_{4n} +\frac{1}{2} s_{2n}.$$ $$\text{ii}) S\ne s.$$

Let $[a_1 , b_1 ] , \ldots, [a_n ,b_n ]$ be a collection of closed intervals such that any of these closed intervals have a point in common. Prove that there exists a point contained in every one of these intervals.

Given the focus $F$ and the directrix $D$ of a parabola $P$ and a line $L$ , describe a euclidean construction for the point or points of intersection of $P$ and $L.$ Be sure to identify the case for which there are no points of intersection.

Let $f(x)$ be a real-valued function, defined for $-1<x<1$ for which $f'(0)$ exists. Let $(a_n) , (b_n)$ be two sequences such that $-1 <a_n <0 <b_n <1$ for all $n$ and $\lim_{n \to \infty } a_n = 0 =\lim_{n \to \infty} b_n.$ Prove that $$\lim_{n \to \infty} \frac{ f(b_n )- f(a_n ) }{b_n -a_n} =f'(0).$$