## THE 1993 ASIAN PACIFIC MATHEMATICAL OLYMPIAD

Time allowed: 4 hours
NO calculators are to be used.
Each question is worth seven points.

Question 1
Let $ABCD$ be a quadrilateral such that all sides have equal length and angle $ABC$ is $60^\circ$.
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 $CA^2 = CM \times CE$.

Question 2
Find the total number of different integer values the function
$f(x)=[x] + [2x]+\left[\frac{5x}{3}\right]+[3x] + [4x]$
takes for real numbers $x$ with $0 = x = 100$.
Question 3
Let
$f(x)= a_nx^n + a_{n-1}x^{n-1} +...+ a_0$ and
•••
$g(x)= c_{n+1}x_{n+1} + c_nx_n +...+ c_0$
•••
be non-zero polynomials with real coeffcients such that $g(x)=(x + r)f(x)$ for some real number $r$. If $a = \max(|a_n|,..., |a_0|)$ and $c = \max(|c_{n+1}|,..., |c_0|)$, prove that $\displaystyle \frac{a}{c} = n + 1$.
Question 4
Determine all positive integers n for which the equation
$x^n + (2+x)^n + (2 - x)^n =0$
has an integer as a solution.
Question 5
Let $P_1, P_2, ..., P_{1993} = P_0$ be distinct points in the $xy$-plane with the following properties:
(i) both coordinates of $P_i$ are integers, for $i =1, 2,..., 1993$;
(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 $i =0, 1,..., 1992$.

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.