## THE 1998 ASIAN PACIFIC MATHEMATICAL OLYMPIAD

Time allowed: 4 hours.
No calculators to be used.
Each question is worth 7 points.

1. Let $F$ be the set of all $n$ -tuples $(A_1, A_2,..., A_n)$ where each $A_i, i=1,2,..., n$ is a subset of ${1,2,...,1998}$. Let $|A|$ denote the number of elements of the set A.
Find the number $\displaystyle \sum_{A_1,A_2,...,A_n}|A_1 \cup A_2 \cup ... \cup A_n|$

2. Show that for any positive integers $a$ and $b$, $(36a + b)(a + 36b)$ cannot be a power of $2$.

3. Let $a, b, c$ be positive real numbers. Prove that
$\displaystyle \left(1+\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right) \geq 2 \left(1+\frac{a+b+c}{\sqrt[3]{abc}}\right)$

4. Let $ABC$ be a triangle and $D$ the foot of the altitude from $A$. Let $E$ and $F$ be on a line through $D$ such that $AE$ is perpendicular to $BE, AF$ is perpendicular to $CF$, and $E$ and $F$ are different from $D$. Let $M$ and $N$ be the midpoints of the line segments $BC$ and $EF$, respectively. Prove that $AN$ is perpendicular to $NM$.

5. Determine the largest of all integers $n$ with the property that $n$ is divisible by all positive integers that are less than $\sqrt[3]{n}$ .