Let $ABCD$ be a quadrilateral such that all sides have equal length and $\angle{ABC} =60^o$. Let $l$ be a line passing through $D$ and not intersecting the quadrilateral (except at $D$). Let $E$ and $F$ be the points of intersection of $l$ with $AB$ and $BC$ respectively. Let $M$ be the point of intersection of $CE$ and $AF$. Prove that $CA^2 = CM \times CE$.

Find the total number of different integer values the function \[ f(x) = [x] + [2x] + [\frac{5x}{3}] + [3x] + [4x] \] takes for real numbers $x$ with $0 \leq x \leq 100$.

Let \begin{eqnarray*} f(x) & = & a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 \ \ \mbox{and} \\ g(x) & = & c_{n+1} x^{n+1} + c_n x^n + \cdots + c_0 \end{eqnarray*} be non-zero polynomials with real coefficients such that $g(x) = (x+r)f(x)$ for some real number $r$. If $a = \max(|a_n|, \ldots, |a_0|)$ and $c = \max(|c_{n+1}|, \ldots, |c_0|)$, prove that $\frac{a}{c} \leq n+1$.

Determine all positive integers $n$ for which the equation \[ x^n + (2+x)^n + (2-x)^n = 0 \] has an integer as a solution.

Let $P_1$, $P_2$, $\ldots$, $P_{1993} = P_0$ be distinct points in the $xy$-plane with the following properties: (i) both coordinates of $P_i$ are integers, for $i = 1, 2, \ldots, 1993$; (ii) there is no point other than $P_i$ and $P_{i+1}$ on the line segment joining $P_i$ with $P_{i+1}$ whose coordinates are both integers, for $i = 0, 1, \ldots, 1992$. Prove that for some $i$, $0 \leq i \leq 1992$, there exists a point $Q$ with coordinates $(q_x, q_y)$ on the line segment joining $P_i$ with $P_{i+1}$ such that both $2q_x$ and $2q_y$ are odd integers.