Question bank APMO 1998

Let $F$ be the set of all $n$-tuples $(A_1,A_2,\ldots,A_n)$ in which every $A_i$ is a subset of $\{1,2,\ldots,1998\}$. Write $|A|$ for the cardinality of a set $A$. Find
$$\sum_{(A_1,\ldots,A_n)\in F}|A_1\cup A_2\cup\cdots\cup A_n|.$$

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

Let $a,b,c$ be positive real numbers. Prove that
$$\left(1+\frac ab\right)\left(1+\frac bc\right)\left(1+\frac ca\right)\ge2\left(1+\frac{a+b+c}{\sqrt[3]{abc}}\right).$$

Let $ABC$ be a triangle and $D$ the foot of the altitude from $A$. Let $E,F$ lie on a line through $D$, with $AE\perp BE$, $AF\perp CF$, and $E,F\ne D$. Let $M,N$ be the midpoints of $BC,EF$, respectively. Prove that $AN\perp NM$.

Determine the largest integer $n$ such that $n$ is divisible by every positive integer less than $\sqrt[3]{n}$.