Putnam 1974

Call a set of positive integers "conspiratorial" if no three of them are pairwise relatively prime. What is the largest number of elements in any "conspiratorial" subset of the integers $1$ to $16$ ?

A circle stands in a plane perpendicular to the ground and a point $A$ lies in this plane exterior to the circle and higher than its bottom. A particle starting from rest at $A$ slides without friction down an inclined straight line until it reaches the circle. Which straight line allows descent in the shortest time?

A well-known theorem asserts that a prime $p > 2$ can be written as the sum of two perfect squares ( $p = m^2 +n^2$ , with $m$ and $n$ integers) if and only if $p \equiv 1$ (mod $4$ ). Assuming this result, find which primes $p > 2$ can be written in each of the following forms, using integers $x$ and $y$ : a) $x^2 +16y^2,$ b) $4x^2 +4xy+ 5y^2.$

An unbiased coin is tossed $n$ times. What is the expected value of $|H-T|$ , where $H$ is the number of heads and $T$ is the number of tails?

Consider the two mutually tangent parabolas $y=x^2$ and $y=-x^2$ . The upper parabola rolls without slipping around the fixed lower parabola. Find the locus of the focus of the moving parabola.

Given $n$ , let $k = k(n)$ be the minimal degree of any monic integral polynomial $$f(x)=x^k + a_{k-1}x^{k-1}+\ldots+a_0$$ such that the value of $f(x)$ is exactly divisible by $n$ for every integer $x.$ Find the relationship between $n$ and $k(n)$ . In particular, find the value of $k(n)$ corresponding to $n = 10^6.$

Which configurations of five (not necessarily distinct) points $p_1 ,\ldots, p_5$ on the circle $x^2 +y^2 =1$ maximize the sum of the ten distances $$\sum_{i<j} d(p_i, p_j)?$$

Let $y(x)$ be a continuously differentiable real-valued function of a real variable $x$ . Show that if $y'(x)^2 +y(x)^3 \to 0$ as $x\to \infty,$ then $y(x)$ and $y'(x) \to 0$ as $x \to \infty.$

Prove that if $a$ is a real number such that $$\cos \pi a= \frac{1}{3},$$ then $a$ is irrational.

A function $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is said to be continuous in each variable separately if, for each fixed value $y_0$ of $y$ , the function $f(x, y_0)$ is contnuous in the usual sense as a function in $x,$ and similarly $f(x_0 , y)$ is continuous as a function of $y$ for each fixed $x_0$ . Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be continuous in each variable separately. Show that there exists a sequence of continuous functions $g_n: \mathbb{R}^{2} \rightarrow \mathbb{R}$ such that $$f(x,y) =\lim_{n\to \infty}g_{n}(x,y)$$ for all $(x,y)\in \mathbb{R}^{2}.$

Show that $$1+\frac{n}{1!} + \frac{n^{2}}{2!} +\ldots+ \frac{n^{n}}{n!} > \frac{e^{n}}{2}$$ for every integer $n\geq 0.$

For a set with $n$ elements, how many subsets are there whose cardinality is respectively $\equiv 0$ (mod $3$ ), $\equiv 1$ (mod $3$ ), $\equiv 2$ (mod $3$ )? In other words, calculate $$s_{i,n}= \sum_{k\equiv i \;(\text{mod} 3)} \binom{n}{k}$$ for $i=0,1,2$ . Your result should be strong enough to permit direct evaluation of the numbers $s_{i,n}$ and to show clearly the relationship of $s_{0,n}, s_{1,n}$ and $s_{2,n}$ to each other for all positive integers $n$ . In particular, show the relationships among these three sums for $n = 1000$ .