SMO 1998

In Fig. , $PA$ and $QB$ are tangents to the circle at $A$ and $B$ respectively. The line $AB$ is extended to meet $PQ$ at $S$ . Suppose that $PA = QB$ . Prove that $QS = SP$ .

Let $N$ be the set of natural numbers, and let $f: N \to N$ be a function satisfying $f(x) + f(x + 2) < 2 f(x + 1)$ for any $x \in N$ . Prove that there exists a straight line in the $xy$ -plane which contains infinitely many points with coordinates $(n,f(n))$ .

Do there exist integers $x$ and $y$ such that $19^{19} = x^3 +y^4$ ? Justify your answer.

Let $n$ be a fixed positive integer. Find all the positive integers $m$ such that $$\frac{m^2+4m}{a_1}+\frac{m^2+8m}{a_1+a_2}+\frac{m^2+12m}{a_1+a_2+a_3}+...+\frac{m^2+4nm}{a_1+a_2+...+a_n}<2500 \left(\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}\right)$$ for any positive numbers $a_1,a_2,...,a_n$ . Justify your answer.