Problem 6 (stars and bars):
If you have $n$ elements and $k$ bars, then you have $\binom{n+1}{k}$ ways to place the bars. If there must be at least one element between the bars, then you can “associate” an element with each bar, so there will be $\binom{n+1-k}{k}$ ways to place the bars.
Problem 8 (geometry):
Occasionally, geometry shapes can be altered without affecting the problem, so in that case always change them to special shapes.
When considering circles like below:
The bisecting line of two lines will pass through all of the centers of these circles.
Also, in cases like these:
Then $\frac{a}{c}=\frac{b}{d}$.
Problem 9 (hyperbola):
Hyperbola equation: $\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$. It has center $(h,k)$. The distance between the vertices (which have coordinates $(h \pm a, k$) is $2a$, the transverse axis. $2b$ is the conjugate axis, and $2c$, the distance between foci, is calculated with $c^2=a^2+b^2$. The equations of the asymptotes are $y-k=\pm \frac{b}{a}(x-h)$.
If a shape is a rhombus, the diagonals bisect each other and are perpendicular.