APMO 1993

Let $A B C D$ be a quadrilateral such that all sides have equal length and angle $\angle A B C$ is 60 degrees. Let $\ell$ be a line passing through $D$ and not intersecting the quadrilateral (except at $D)$. Let $E$ and $F$ be the points of intersection of $\ell$ with $A B$ and $B C$ respectively. Let $M$ be the point of intersection of $C E$ and $A F$. Prove that $C A^{2}=C M \times C E$.

Find the total number of different integer values the function

$$f(x)=[x]+[2 x]+\left[\frac{5 x}{3}\right]+[3 x]+[4 x]$$

takes for real numbers $x$ with $0 \leq x \leq 100$. Note: $[t]$ is the largest integer that does not exceed $t$.

Let

$$f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0} \text { and } g(x)=c_{n+1} x^{n+1}+c_{n} x^{n}+\cdots+c_{0}$$

be non-zero polynomials with real coefficients such that $g(x)=(x+r) f(x)$ for some real number $r$. If $a=\max \left(\left|a_{n}\right|, \ldots,\left|a_{0}\right|\right)$ and $c=\max \left(\left|c_{n+1}\right|, \ldots,\left|c_{0}\right|\right)$, 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 $x y$-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 $\left(q_{x}, q_{y}\right)$ on the line segment joining $P_{i}$ with $P_{i+1}$ such that both $2 q_{x}$ and $2 q_{y}$ are odd integers.