Putnam 1979

Problem A-1 Find positive integers $n$ and $a_{1}, a_{2}, \ldots, a_{n}$ such that $$a_{1}+a_{2}+\cdots+a_{n}=1979$$ and the product $a_{1} a_{2} \cdots a_{n}$ is as large as possible.

Problem A-2 Establish necessary and sufficient conditions on the constant $\boldsymbol{k}$ for the existence of a continuous real valued function $f(x)$ satisfying $f(f(x))=k x^{9}$ for all real $x$.

Problem A-3 Let $x_{1}, x_{2}, x_{3}, \ldots$ be a sequence of nonzero real numbers satisfying $$x_{n}=\frac{x_{n-2} x_{n-1}}{2 x_{n-2}-x_{n-1}} \text { for } n=3,4,5, \ldots$$

Establish necessary and sufficient conditions on $x_{1}$ and $x_{2}$ for $x_{n}$ to be an integer for infinitely many values of $n$.

Problem A-4 Let $A$ be a set of $2 n$ points in the plane, no three of which are collinear. Suppose that $n$ of them are colored red and the remaining $n$ blue. Prove or disprove: there are $n$ closed straight line segments, no two with a point in common, such that the endpoints of each segment are points of $A$ having different colors.

Problem A-5 Denote by $[x]$ the greatest integer less than or equal to $x$ and by $S(x)$ the sequence $[x],[2 x],[3 x], \ldots$. Prove that there are distinct real solutions $\alpha$ and $\beta$ of the equation $x^{3}-10 x^{2}+29 x-25=0$ such that infinitely many positive integers appear both in $S(\alpha)$ and in $S(\beta)$.

Problem A-6 Let $0<p_{i}<1$ for $i=1,2, \ldots, n$. Show that $$\sum_{i=1}^{n} \frac{1}{\left|x-p_{i}\right|}<8 n\left(1+\frac{1}{3}+\frac{1}{5}+\cdots+\frac{1}{2 n-1}\right)$$ for some $x$ satisfying $0<x<1$.

Problem B-1 Prove or disprove: there is at least one straight line normal to the graph of $y=\cosh x$ at a point $(a, \cosh a)$ and also normal to the graph of $y=\sinh x$ at a point $(c, \sinh c)$. [At a point on a graph, the normal line is the perpendicular to the tangent at that point. Also, $\cosh x=\left(e^{x}+\right.$ $\left.e^{-x}\right) / 2$ and $\left.\sinh x=\left(e^{x}-e^{-x}\right) / 2.\right]$

Problem B-2 Let $0<a<b$. Evaluate $$\lim _{t \rightarrow 0}\left\{\int_{0}^{1}[b x+a(1-x)]^{\prime} d x\right\}^{1 / t}$$ [The final answer should not involve any operations other than addition, subtraction, multiplication, division, and exponentiation.]

Problem B-3 Let $F$ be a finite field having an odd number $m$ of elements. Let $p(x)$ be an irreducible (i.e., nonfactorable) polynomial over $F$ of the form $$x^{2}+b x+c, b, c \in F$$

For how many elements $k$ in $F$ is $p(x)+k$ irreducible over $F$ ?

Problem B-4 (a) Find a solution that is not identically zero, of the homogeneous linear differential equation $$\left(3 x^{2}+x-1\right) y^{\prime \prime}-\left(9 x^{2}+9 x-2\right) y^{\prime}+(18 x+3) y=0$$

Intelligent guessing of the form of a solution may be helpful. (b) Let $y=f(x)$ be the solution of the nonhomogeneous differential equation $$\left(3 x^{2}+x-1\right) y^{\prime \prime}-\left(9 x^{2}+9 x-2\right) y^{\prime}+(18 x+3) y=6(6 x+1)$$ that has $f(0)=1$ and $(f(-1)-2)(f(1)-6)=1$. Find integers $a, b, c$ such that $(f(-2)-a)(f(2)-b)=c$.

Problem B-5 In the plane, let $C$ be a closed convex set that contains $(0,0)$ but no other point with integer coordinates. Suppose that $A(C)$, the area of $C$, is equally distributed among the four quadrants. Prove that $A(C)<4$.

Problem B-6 For $k=1,2, \ldots, n$ let $z_{k}=x_{k}+i y_{k}$, where the $x_{k}$ and $y_{k}$ are real and $i=\sqrt{-1}$. Let $r$ be the absolute value of the real part of $$\pm \sqrt{z_{1}^{2}+z_{2}^{2}+\cdots+z_{n}^{2}}$$

Prove that $r<\left|x_{1}\right|+\left|x_{2}\right|+\cdots+\left|x_{n}\right|$.