There are $1001$ points in the plane such that no three are collinear. The points are joined by $1001$ line segments such that each point is an endpoint of exactly two of the line segments. Prove that there does not exist a straight line in the plane that intersects each of the $1001$ segments in an interior point. An interior point of a line segment is a point of the line segment that is not one of the two endpoints.

Fiona has a deck of cards labelled $1$ to $n$, laid out in a row on the table in order from $1$ to $n$ from left to right. Her goal is to arrange them in a single pile, through a series of steps of the following form: If at some stage the cards are in $m$ piles, she chooses $1\leq k<m$ and arranges the cards into $k$ piles by picking up pile $k+1$ and putting it on pile $1$; picking up pile $k+2$ and putting it on pile $2$; and so on, working from left to right and cycling back through as necessary. She repeats the process until the cards are in a single pile, and then stops. So for example, if $n=7$ and she chooses $k=3$ at the first step she would have the following three piles: $ \begin{matrix} 7 & \ &\ \\ 4 & 5 & 6 \\ 1 &2 & 3 \\ \hline \end{matrix} $ If she then chooses $k=1$ at the second stop, she finishes with the cards in a single pile with cards ordered $6352741$ from top to bottom. How many different final piles can Fiona end up with?

Determine the set of real numbers $\alpha$ that can be expressed in the form \[\alpha=\sum_{n=0}^{\infty}\frac{x_{n+1}}{x_n^3}\]where $x_0,x_1,x_2,\dots$ is an increasing sequence of real numbers with $x_0=1$.

A regular spatial pentagon consists of five points $P_1,P_2,P_3,P_4$ and $P_5$ in $\mathbb{R}^3$ such that $|P_iP_{i+1}|=|P_jP_{j+1}|$ and $\angle P_{i-1}P_iP_{i+1}=\angle P_{j-1}P_jP_{j+1}$ for all $1\leq i,\leq 5$, where $P_0=P_5$ and $P_{6}=P_{1}$. A regular spatial pentagon is planar if there is a plane passing through all five points $P_1,P_2,P_3,P_4$ and $P_5$. Show that every regular spatial pentagon is planar.

Let $\mathcal{M}$ be the set of $5\times 5$ real matrices of rank $3$. Given a matrix in $\mathcal{M}$, the set of columns of $A$ has $2^5-1=31$ nonempty subsets. Let $k_A$ be the number of these subsets that are linearly independent. Determine the maximum and minimum values of $k_A$, as $A$ varies over $\mathcal{M}$. The rank of a matrix is the dimension of the span of its columns.

For each positive integer $k$, let $S_k$ be the set of real numbers that can be expressed in the form \[\frac{1}{n_1}+\frac{1}{n_2}+\dots+\frac{1}{n_k},\]where $n_1,n_2\dots,n_k$ are positive integers. Prove that $S_k$ does not contain an infinite strictly increasing sequence.

A cat is trying to catch a mouse in the non-negative quadrant \[N=\{(x_1,x_2)\in \mathbb{R}^2: x_1,x_2\geq 0\}.\]At time $t=0$ the cat is at $(1,1)$ and the mouse is at $(0,0)$. The cat moves with speed $\sqrt{2}$ such that the position $c(t)=(c_1(t),c_2(t))$ is continuous, and differentiable except at finitely many points; while the mouse moves with speed $1$ such that its position $m(t)=(m_1(t),m_2(t))$ is also continuous, and differentiable except at finitely many points. Thus $c(0)=(1,1)$ and $m(0)=(0,0)$; $c(t)$ and $m(t)$ are continuous functions of $t$ such that $c(t),m(t)\in N$ for all $t\geq 0$; the derivatives $c'(t)=(c'_1(t),c'_2(t))$ and $m'(t)=(m'_1(t),m'_2(t))$ each exist for all but finitely many $t$ and \[(c'_1(t)^2+(c'_2(t))^2=2 \qquad (m'_1(t)^2+(m'_2(t))^2=1,\]whenever the respective derivative exists. At each time $t$ the cat knows both the mouse's position $m(t)$ and velocity $m'(t)$. Show that, no matter how the mouse moves, the cat can catch it by time $t=1$; that is, show that the cat can move such that $c(\tau)=m(\tau)$ for some $\tau\in[0,1]$.

The following problem is open in the sense that no solution is currently known to part (b). Let $n\geq 2$ be an integer, and $P_n$ be a regular polygon with $n^2-n+1$ vertices. We say that $n$ is $\emph{taut}$ if it is possible to choose $n$ of the vertices of $P_n$ such that the pairwise distances between the chosen vertices are all distinct. (a) show that if $n-1$ is prime then $n$ is taut. (b) Which integers $n\geq 2$ are taut?