HMIC 2013

Let $\mathcal{S}$ be a set of size $n$, and $k$ be a positive integer. For each $1 \leq i \leq k n$, there is a subset $S_{i} \subset \mathcal{S}$ such that $\left|S_{i}\right|=2$. Furthermore, for each $e \in \mathcal{S}$, there are exactly $2 k$ values of $i$ such that $e \in S_{i}$. Show that it is possible to choose one element from $S_{i}$ for each $1 \leq i \leq k n$ such that every element of $\mathcal{S}$ is chosen exactly $k$ times.

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that, for all real numbers $x, y$,

$$(x-y)(f(x)-f(y))=f(x-f(y)) f(f(x)-y)$$

Triangle $A B C$ is inscribed in a circle $\omega$ such that $\angle A=60^{\circ}$ and $\angle B=75^{\circ}$. Let the bisector of angle $A$ meet $B C$ and $\omega$ at $E$ and $D$, respectively. Let the reflections of $A$ across $D$ and $C$ be $D^{\prime}$ and $C^{\prime}$, respectively. If the tangent to $\omega$ at $A$ meets line $B C$ at $P$, and the circumcircle of $A P D^{\prime}$ meets line $A C$ at $F \neq A$, prove that the circumcircle of $C^{\prime} F E$ is tangent to $B C$ at $E$.

A subset $U \subset \mathbb{R}$ is open if for any $x \in U$, there exist real numbers $a, b$ such that $x \in(a, b) \subset U$. Suppose $S \subset \mathbb{R}$ has the property that any open set intersecting $(0,1)$ also intersects $S$. Let $T$ be a countable collection of open sets containing $S$. Prove that the intersection of all of the sets of $T$ is not a countable subset of $\mathbb{R}$. (A set $\Gamma$ is countable if there exists a bijective function $f: \Gamma \rightarrow \mathbb{Z}$.)

(a) Given a finite set $X$ of points in the plane, let $f_{X}(n)$ be the largest possible area of a polygon with at most $n$ vertices, all of which are points of $X$. Prove that if $m, n$ are integers with $m \geq n>2$, then $f_{X}(m)+f_{X}(n) \geq f_{X}(m+1)+f_{X}(n-1)$. (b) Let $P_{0}$ be a 1-by-2 rectangle (including its interior), and inductively define the polygon $P_{i}$ to be the result of folding $P_{i-1}$ over some line that cuts $P_{i-1}$ into two connected parts. The diameter of a polygon $P_{i}$ is the maximum distance between two points of $P_{i}$. Determine the smallest possible diameter of $P_{2013}$. (In other words, given a polygon $P_{i-1}$, a fold of $P_{i-1}$ consists of a line $l$ dividing $P_{i-1}$ into two connected parts $A$ and $B$, and the folded polygon $P_{i}=A \cup B_{l}$, where $B_{l}$ is the reflection of $B$ over the line l.)