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\homework{17: Putnam Problems}{Due Tuesday, Week 10, at the start of class.}


Solve \textbf{three} of the following \textbf{six} problems.  All problems here are taken from past Putnam exams.    As always, prove your claims/have fun!  

\begin{enumerate}
\item[0.] Solve any un-signed-up-for problems from HW\#16!
\item Take any positive nonprime integer $n > 1$.  Show that we can find three positive integers $a,b,c$ such that
\begin{align*}
n = ab + ac + bc + 1.
\end{align*}
\item Take the unit sphere $S = \{(x,y,z) ~|~ x^2 + y^2 + z^2 = 1\}$.  Pick five points $A,B,C,D,E$ on this sphere.  Show that there is some way to cut this sphere in half, so that at least four of these points are on the same half of our sphere.
\item Suppose that $P,Q,R$ are three points in the plane with the following properties:
\begin{itemize}
\item $P, Q, R$ have integer co\"ordinates.
\item The three points $P, Q, R$ are not collinear: that is, there is no single line $L$ that contains all three of these points.
\item The distances $d(P,Q), d(P,R), d(Q,R)$ are all integers.
\end{itemize}
What is the smallest possible value for the distance $d(P,Q)$?
\item Prove or disprove: there is no sequence $\{x_k\}_{k=1}^\infty$ of real numbers such that for every $n \in \mathbb{N}$, we have
\begin{align*}
\sum_{k=1}^\infty x_k^n = n.
\end{align*}
\item Suppose that there is a platform off the coasts of the Catalina Islands outfitted with sensors that measure the tides\footnote{ The tide $T_t$ at any time $t$, for this problem, is simply the distance from the sea floor to the platform.}.  Tides are fairly predictable, and over time scientists have noticed that if $t$ is the current time and $T_t$ is the height of the tides at time $t$, there is some fixed polynomial of degree at most 3, $p(x)$ such that for any $t$, we have $T_t = p(t).$

However, scientists have not yet determined what $p(x)$ is, and are facing budget cuts that will stop them from being able to continuously measure the tides at this platform.
\begin{enumerate}
\item Show that without knowing what $p(x)$ is, we can find two times $t_1 < t_2$ such that the average tidal height from 9am to 3pm is $\frac{p(t_1) + p(t_2)}{2}$.
\item Show that $t_1 \approx $10:15am, $t_2 \approx$ 1:45pm.
\end{enumerate} 
\item Find all polynomials $p(x)$ such that $f(0) = 0$ and for all $x$, $f(x^2 + 1) = (f(x))^2 + 1$.
\end{enumerate}





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