Putnam 1960

Let $n$ be a given positive integer. How many solutions are there in ordered positive integer pairs $(x,y)$ to the equation $$\frac{xy}{x+y}=n?$$

Show that if three points are inside are closed square of unit side, then some pair of them are within $\sqrt{6}-\sqrt{2}$ units apart.

Show that if $t_1 , t_2, t_3, t_4, t_5$ are real numbers, then $$\sum_{j=1}^{5} (1-t_j )\exp \left( \sum_{k=1}^{j} t_k \right) \leq e^{e^{e^{e}}}.$$

Given two points, $P$ and $Q$ , on the same side of a line $L$ , the problem is to find a third point $R$ so that $PR+ RQ+RS$ is minimal, where $S$ is the unique point on $L$ such that $RS$ is perpendicular to $L.$ Consider all cases.

Find all polynomials $f(x)$ with real coefficients having the property $f(g(x))=g(f(x))$ for every polynomial $g(x)$ with real coefficients.

A player repeatedly throwing a die is to play until their score reaches or passes a total $n$ . Denote by $p(n)$ the probability of making exactly the total $n,$ and find the value of $\lim_{n \to \infty} p(n).$

Find all integer solutions $(m,n)$ to $m^{n}=n^{m}.$

Evaluate the double series $$\sum_{j=0}^{\infty} \sum_{k=0}^{\infty} 2^{-3k -j -(k+j)^{2}}.$$

The motion of the particles of a fluid in the plane is specified by the following components of velocity $$\frac{dx}{dt}=y+2x(1-x^2 -y^2),\; \frac{dy}{dt}=-x.$$ Sketch the shape of the trajectories near the origin. Discuss what happens to an individual particle as $t\to \infty$ , and justify your conclusion.

Consider the arithmetic progression $a, a+d, a+2d,\ldots$ where $a$ and $d$ are positive integers. For any positive integer $k$ , prove that the progression has either no $k$ -th powers or infinitely many.

Define a sequence $(a_n)$ by $a_0 =0$ and $a_n = 1 +\sin(a_{n-1}-1)$ for $n\geq 1$ . Evaluate $$\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^{n} a_k.$$

Any positive integer $n$ can be written in the form $n=2^{k}(2l+1)$ with $k,l$ positive integers. Let $a_n =e^{-k}$ and $b_n = a_1 a_2 a_3 \cdots a_n.$ Prove that $$\sum_{n=1}^{\infty} b_n$$ converges.