Putnam 1959

Let $n$ be a positive integer. Prove that $x^n -\frac{1}{x^{n}}$ is expressible as a polynomial in $x-\frac{1}{x}$ with real coefficients if and only if $n$ is odd.

"Let $\omega^3 = 1, \omega \neq 1$ . Show that $z_1, z_2, -\omega z_1, -\omega^2z_2$ are the vertices of an equilateral triangle."

Find all complex-valued functions $f$ of a complex variable such that $$f(z)+zf(1-z)=1+z$$ for all $z\in \mathbb{C}$ .

If $f$ and $g$ are real-valued functions of one real variable, show that there exist $x$ and $y$ in $[0,1]$ such that $$|xy-f(x)-g(y)|\geq \frac{1}{4}.$$

A sparrow, flying horizontally in a straight line, is $50$ feet directly below an eagle and $100$ feet directly above a hawk. Both hawk and eagle fly directly toward the sparrow, reaching it simultaneously. The hawk flies twice as fast as the sparrow. How far does each bird fly? At what rate does the eagle fly?

Let $m$ and $n$ be integers greater than $1$ and $a_1 ,a_2 ,\ldots, a_{m+1}$ be real numbers. Prove that there exist real $n\times n$ matrices $A_1 ,A_2,\ldots, A_m$ such that (i) $\det(A_j) =a_j$ for $j=1,2,\ldots,m$ and (ii) $\det(A_1 +A_2 +\ldots+A_m)=a_{m+1}.$

Let each of $m$ distinct points on the positive part of the $x$ -axis be joined to $n$ distinct points on the positive part of the $y$ -axis. Obtain a formula for the number of intersection points of these segments, assuming that no three of the segments are concurrent.

Let $c$ be a positive real number. Prove that $c$ can be expressed in infinitely many ways as a sum of infinitely many distinct terms selected from the sequence $\left( \frac{1}{10n} \right)_{n\in \mathbb{N}}$

Give an example of a continuous real-valued function $f$ form $[0,1]$ to $[0,1]$ which takes on every value in $[0,1]$ an infinite number of times.

Given the following matrix $$\begin{pmatrix} 11& 17 & 25& 19& 16 24 &10 &13 & 15&3 12 &5 &14& 2&18 23 &4 &1 &8 &22 6&20&7 &21&9 \end{pmatrix},$$ choose five of these elements, no two from the same row or column, in such a way that the minimum of these elements is as large as possible.

Find the equation of the smallest sphere which is tangent to both of the lines $$\begin{pmatrix} xyz \end{pmatrix} =\begin{pmatrix} t+1 2t+4 -3t +5 \end{pmatrix},\;\;\;\begin{pmatrix} xyz \end{pmatrix} =\begin{pmatrix} 4t-12 -t+8 t+17 \end{pmatrix}.$$