# Morning Session ## Problem 1 Let $n$ be a given positive integer. How many solutions are there in ordered positive integer pairs $(x, y)$ to the equation $\frac{x y}{x+y}=n$? ## Problem 2 Show that if three points are inside a closed square of unit side, then some pair of them are within $\sqrt{6}-\sqrt{2}$ units apart. ## Problem 3 Show that if $t_1, t_2, t_3, t_4, t_5$ are real numbers, then $\sum_{j=1}^5\left(1-t_j\right) \exp \left(\sum_{k=1}^j t_k\right) \leq e^{e^{e^e}} .$ ## Problem 4 Given two points in the plane, $P$ and $Q$, at fixed distances from a line $L$, and on the same side of the line, as indicated, the problem is to find a third point $R$ so that $P R+R Q+R S$ is a minimum, where $R S$ is perpendicular to $L$. Consider all cases. ![[1960-12-03 Putnam-diagramP4.svg]] ## Problem 5 Consider a polynomial $f(x)$ with real coefficients having the property $f(g(x))=g(f(x))$ for every polynomial $g(x)$ with real coefficients. Determine and prove the nature of $f(x)$. ## Problem 6 A player throwing a die scores as many points as on the top face of the die and is to play until his score reaches or passes a total $n$. Denote by $p(n)$ the probability of making exactly the total $n$, and find the value of $\lim _{n-\infty} p(n)$. ## Problem 7 Let $N(n)$ denote the smallest positive integer $N$ such that $x^N=1$ for every permutation $x$ on $n$ symbols, where 1 denotes the identity permutation. Prove that if $n>1$, $ \begin{aligned} \frac{N(n)}{N(n-1)} & =1 \text { if } n \text { is divisible by } 2 \text { distinct primes, } \\ & =p \text { if } n \text { is a power of a prime } p. \end{aligned} $ # Afternoon Session ## Problem 1 Find all solutions of $n^m=m^n$ in integers $n$ and $m(x \neq m)$. Prove that you have obtained all of them. ## Problem 2 Evaluate the double series $ \sum_{j=0}^{\infty} \sum_{k=0}^{\infty} 2^{-3 k-j-(k+j)^2} . $ ## Problem 3 The motion of the particles of a fluid in the plane is specified by the following components of velocity (1) $ \frac{d x}{d t}=y+2 x\left(1-x^2-y^2\right) $ $ \frac{d y}{d t}=-x . $ Sketch the shape of the trajectories near the origin. Discuss what happens to an individual particle as $t-+\infty$, and justify your conclusion. Note. The second $y$ in the first equation above was broken and looked like a $v$ on the examination as it was distributed to the contestants. ## Problem 4 Consider the arithmetic progression $a$, $a+d$, $a+2 d$, ..., where $a$ and $d$ are positive integers. For any positive integer $k$, prove that the progression has either no exact $k^{\text{th}}$ powers or infinitely many. ## Problem 5 Define a sequence as follows: $ \begin{aligned} & a_0=0 \\ & a_1=1+\sin (-1) \\ & \ \vdots \\ & a_n=1+\sin \left(a_{n-1}-1\right) \\ & \ \vdots \end{aligned} $ Evaluate $ \lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^n a_k $ ## Problem 6 Any positive integer may be written in the form $n=2^k(2 l+1)$. Let $a_n$ $=e^{-k}$ and $b_n=a_1 a_2 a_3 \cdots a_n$. Prove that $\Sigma b_n$ converges. ## Problem 7 Let $g(t)$ and $h(t)$ be real, continuous functions for $t \geq 0$. Show that any function $v(t)$ satisfying the differential inequality $ \frac{d v}{d t}+g(t) v \geq h(t), \quad v(0)=c, $ satisfies the further inequality $v(t) \geq u(t)$ where $ \frac{d u}{d t}+g(t) u=h(t), \quad u(0)=c . $ From this, conclude that for sufficiently small $t>0$, the solution of $ \frac{d v}{d t}+g(t) v=v^2, \quad v(0)=c_1 $ may be written $ v=\max _w\left[c_1 e^{-\int_0^t[g(s)-2 w(s)] d s}-\int_0^t e^{-\int_{0}^t \left[g\left(s_1\right)-2 w\left(s_1\right)\right] d s_1}w^2(s) d s\right] $ where the maximization is over all continuous functions $w(t)$ defined over some $t$-interval $\left[0, t_0\right]$.