APMO 1998

Let $F$ be the set of all $n$ -tuples $(A_1, \ldots, A_n)$ such that each $A_{i}$ is a subset of $\{1, 2, \ldots, 1998\}$ . Let $|A|$ denote the number of elements of the 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 $a$ and $b$ , $(36a+b)(a+36b)$ cannot be a power of $2$ .

Let $a$ , $b$ , $c$ be positive real numbers. Prove that $$\biggl(1+\frac{a}{b}\biggr) \biggl(1+\frac{b}{c}\biggr) \biggl(1+\frac{c}{a}\biggr) \ge 2 \biggl(1+\frac{a+b+c}{\sqrt[3]{abc}}\biggr).$$

Let $ABC$ be a triangle and $D$ the foot of the altitude from $A$ . Let $E$ and $F$ lie on a line passing 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 segments $BC$ and $EF$ , respectively. Prove that $AN$ is perpendicular to $NM$ .

What is the largest integer divisible by all positive integers less than its cube root.