[b]p1.[/b] I open my $2010$-page dictionary, whose pages are numbered $ 1$ to $2010$ starting on page $ 1$ on the right side of the spine when opened, and ending with page $2010$ on the left. If I open to a random page, what is the probability that the two page numbers showing sum to a multiple of $6$?


[b]p2.[/b] Let $A$ be the number of positive integer factors of $128$.
Let $B$ be the sum of the distinct prime factors of $135$.
Let $C$ be the units’ digit of $381$.
Let $D$ be the number of zeroes at the end of $2^5\cdot 3^4 \cdot 5^3 \cdot 7^2\cdot 11^1$.
Let $E$ be the largest prime factor of $999$.
Compute $\sqrt[3]{\sqrt{A + B} +\sqrt[3]{D^C+E}}$.


[b]p3. [/b] The root mean square of a set of real numbers is defined to be the square root of the average of the squares of the numbers in the set. Determine the root mean square of $17$ and $7$.


[b]p4.[/b] A regular hexagon $ABCDEF$ has area $1$. The sides$ AB$, $CD$, and $EF$ are extended to form a larger polygon with $ABCDEF$ in the interior. Find the area of this larger polygon.


[b]p5.[/b] For real numbers $x$, let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$. For example, $\lfloor 3\rfloor  = 3$ and $\lfloor 5.2 \rfloor = 5$. Evaluate $\lfloor -2.5 \rfloor + \lfloor \sqrt 2 \rfloor + \lfloor -\sqrt 2 \rfloor + \lfloor 2.5 \rfloor$.


[b]p6.[/b] The mean of five positive integers is $7$, the median is $8$, and the unique mode is $9$. How many possible sets of integers could this describe?


[b]p7.[/b] How many three digit numbers x are there such that $x + 1$ is divisible by $11$?


[b]p8.[/b] Rectangle $ABCD$ is such that $AD = 10$ and $AB > 10$. Semicircles are drawn with diameters $AD$ and $BC$ such that the semicircles lie completely inside rectangle $ABCD$. If the area of the region inside $ABCD$ but outside both semicircles is $100$, determine the shortest possible distance between a point $X$ on semicircle $AD$ and $Y$ on semicircle $BC$.


[b]p9.[/b] $ 8$ distinct points are in the plane such that five of them lie on a line $\ell$, and the other three points lie off the line, in a way such that if some three of the eight points lie on a line, they lie on $\ell$. How many triangles can be formed using some three of the $ 8$ points?


[b]p10.[/b] Carl has $10$ Art of Problem Solving books, all exactly the same size, but only $9$ spaces in his bookshelf. At the beginning, there are $9$ books in his bookshelf, ordered in the following way.
$A - B - C - D - E - F - G - H - I$
He is holding the tenth book, $J$, in his hand. He takes the books out one-by-one, replacing each with the book currently in his hand. For example, he could take out $A$, put $J$ in its place, then take out $D$, put $A$ in its place, etc. He never takes the same book out twice, and stops once he has taken out the tenth book, which is $G$. At the end, he is holding G in his hand, and his bookshelf looks like this.
$C - I - H - J - F - B - E - D - A$
Give the order (start to finish) in which Carl took out the books, expressed as a $9$-letter string (word).


PS. You had better use hide for answers.
1. To determine the number of triangles that can be formed using some three of the 8 points, we start by calculating the total number of ways to choose 3 points out of 8. This is given by the binomial coefficient:
   \[
   \binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56
   \]
2. Next, we need to subtract the number of degenerate triangles (i.e., sets of 3 collinear points) from the total. Since 5 of the points lie on a line \( \ell \), the number of ways to choose 3 points from these 5 points is:
   \[
   \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10
   \]
3. Therefore, the number of non-degenerate triangles (i.e., triangles that are not collinear) is:
   \[
   56 - 10 = 46
   \]

The final answer is \(\boxed{46}\).