Putnam 1977

Problem A-1 Consider all lines which meet the graph of $$y=2 x^{4}+7 x^{3}+3 x-5$$ in four distinct points, say $\left(x_{i}, y_{i}\right), i=1,2,3,4$. Show that $$\frac{x_{1}+x_{2}+x_{3}+x_{4}}{4}$$ is independent of the line and find its value.

Problem A-2 Determine all solutions in real numbers $x, y, z, w$ of the system $$\begin{aligned} x+y+z & =w \\ \frac{1}{x}+\frac{1}{y}+\frac{1}{z} & =\frac{1}{w} \end{aligned}$$

Problem A-3 Let $u, f$, and $g$ be functions, defined for all real numbers $x$, such that $$\frac{u(x+1)+u(x-1)}{2}=f(x) \text { and } \frac{u(x+4)+u(x-4)}{2}=g(x) .$$

Determine $u(x)$ in terms of $f$ and $g$.

Problem A-4 For $0<x<1$, express $$\sum_{n=0}^{\infty} \frac{x^{2^{n}}}{1-x^{2^{n+1}}}$$ as a rational function of $x$.

Problem A-5 Prove that $$\binom{p a}{p b} \equiv\binom{a}{b}(\bmod p)$$ for all integers $p, a$, and $b$ with $p$ a prime, $p>0$, and $a>b>0$. Notation: $\binom{m}{n}$ denotes the binomial coefficient $\frac{m!}{n!(m-n)!}$.

Problem A-6 Let $f(x, y)$ be a continuous function on the square $$S=\{(x, y): 0<x<1,0<y<1\} .$$

For each point $(a, b)$ in the interior of $S$, let $S_{(a, b)}$ be the largest square that is contained in $S$, is centered at $(a, b)$, and has sides parallel to those of $S$. If the double integral $\iint f(x, y) d x d y$ is zero when taken over each square $S_{(a, b)}$ must $f(x, y)$ be identically zero on $S$ ?

Problem B-1 Evaluate the infinite product $$\prod_{n=2}^{\infty} \frac{n^{3}-1}{n^{3}+1}$$

Problem B-2 Given a convex quadrilateral $A B C D$ and a point $O$ not in the plane of $A B C D$, locate point $A^{\prime}$ on line $O A$, point $B^{\prime}$ on line $O B$, point $C^{\prime}$ on line $O C$, and point $D^{\prime}$ on line $O D$ so that $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ is a parallelogram.

Problem B-3 An (ordered) triple ( $x_{1}, x_{2}, x_{3}$ ) of positive irrational numbers with $x_{1}+x_{2}+x_{3}=1$ is called "balanced" if each $x_{i}<1 / 2$. If a triple is not balanced, say if $x_{j}>1 / 2$, one performs the following "balancing act" $$B\left(x_{1}, x_{2}, x_{3}\right)=\left(x_{1}^{\prime}, x_{2}^{\prime}, x_{3}^{\prime}\right),$$ where $x_{i}^{\prime}=2 x_{i}$ if $i \neq j$ and $x_{j}^{\prime}=2 x_{j}-1$. If the new triple is not balanced, one performs the balancing act on it. Does continuation of this process always lead to a balanced triple after a finite number of performances of the balancing act?

Problem B-4 Let $C$ be a continuous closed curve in the plane which does not cross itself and let $Q$ be a point inside $C$. Show that there exist points $P_{1}$ and $P_{2}$ on $C$ such that $Q$ is the midpoint of the line segment $P_{1} P_{2}$.

Problem B-5 Suppose that $a_{1}, a_{2}, \ldots, a_{n}$ are real $(n>1)$ and $$A+\sum_{i=1}^{n} a_{i}^{2}<\frac{1}{n-1}\left(\sum_{i=1}^{n} a_{i}\right)^{2}$$

Prove that $A<2 a_{i} a_{j}$ for $1<i<j<n$.

Problem B-6 Let $H$ be a subgroup with $h$ elements in a group $G$. Suppose that $G$ has an element $a$ such that for all $x$ in $H$, $(x a)^{3}=1$, the identity. In $G$, let $P$ be the subset of all products $x_{1} a x_{2} a \cdots x_{n} a$, with $n$ a positive integer and the $x_{i}$ in H. (a) Show that $P$ is a finite set. (b) Show that, in fact, $P$ has no more than $3 h^{2}$ elements.