Let $ \prod_{n=1}^{1996}{(1+nx^{3^n})}= 1+ a_{1}x^{k_{1}}+ a_{2}x^{k_{2}}+...+ a_{m}x^{k_{m}}$ where $a_{1}, a_{1}, . . . , a_{m}$ are nonzero and $k_{1} < k_{2} <...< k_{m}$. Find $a_{1996}$.

In a parallelogram $ABCD$ with $\angle A < 90$, the circle with diameter $AC$ intersects the lines $CB$ and $CD$ again at $E$ and $F$ , and the tangent to this circle at $A$ meets the line $BD$ at $P$ . Prove that the points $P$, $E$, $F$ are collinear.

If $0=x_{1}<x_{2}<...<x_{2n+1}=1$ are real numbers with $x_{i+1}-x_{i} \leq h$ for $1 \leq i \leq 2n$, show that $\frac{1-h}{2}<\sum_{i=1}^{n}{x_{2i}(x_{2i+1}-x_{2i-1})}\leq \frac{1+h}{2}$

The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ with $S_{ABC} = S_{ADC}$ intersect at $E$. The lines through $E$ parallel to $AD$, $DC$, $CB$, $BA$ meet $AB$, $BC$, $CD$, $DA$ at $K$, $L$, $M$, $N$, respectively. Compute the ratio $\frac{S_{KLMN}}{S_{ABC}}$

Find the maximum number of pairwise disjoint sets of the form $S_{a,b} = \{n^{2}+an+b | n \in \mathbb{Z}\}$, $a, b \in \mathbb{Z}$.

Determine all ordered pairs of positive real numbers $(a, b)$ such that every sequence $(x_{n})$ satisfying $\lim_{n \rightarrow \infty}{(ax_{n+1} - bx_{n})} = 0$ must have $\lim_{n \rightarrow \infty} x_n = 0$.