[b][b]p1.[/b][/b] Find the perimeter of a regular hexagon with apothem $3$.


[b]p2.[/b] Concentric circles of radius $1$ and r are drawn on a circular dartboard of radius $5$. The probability that a randomly thrown dart lands between the two circles is $0.12$. Find $r$.


[b]p3.[/b] Find all ordered pairs of integers $(x, y)$ with $0 \le x \le 100$,  $0 \le y \le 100$ satisfying $$xy = (x - 22) (y + 15) .$$


[b]p4.[/b] Points $A_1$,$A_2$,$...$,$A_{12}$ are evenly spaced around a circle of radius $1$, but not necessarily in order. Given that chords $A_1A_2$, $A_3A_4$, and $A_5A_6$ have length $2$ and chords $A_7A_8$ and $A_9A_{10}$ have length $2 sin (\pi / 12)$, find all possible lengths for chord $A_{11}A_{12}$.


[b]p5.[/b] Let $a$ be the number of digits of $2^{1998}$, and let $b$ be the number of digits in $5^{1998}$. Find $a + b$.


[b]p6.[/b] Find the volume of the solid in $R^3$ defined by the equations
$$x^2 + y^2 \le 2$$
$$x + y + |z| \le 3.$$


[b]p7.[/b] Positive integer $n$ is such that $3n$ has $28$ positive divisors and $4n$ has $36$ positive divisors. Find the number of positive divisors of $n$.


[b]p8.[/b] Define functions $f$ and $g$ by $f (x) = x +\sqrt{x}$ and $g (x) = x + 1/4$. Compute $$g(f(g(f(g(f(g(f(3)))))))).$$
(Your answer must be in the form $a + b \sqrt{ c}$ where $a$, $b$, and $c$ are rational.)


[b]p9.[/b] Sequence $(a_1, a_2,...)$ is defined recursively by $a_1 = 0$, $a_2 = 100$, and $a_n = 2a_{n-1}-a_{n-2}-3$. Find the greatest term in the sequence $(a_1, a_2,...)$.


[b]p10.[/b] Points $X = (3/5, 0)$ and $Y = (0, 4/5)$ are located on a Cartesian coordinate system. Consider all line segments which (like $\overline{XY}$ ) are of length 1 and have one endpoint on each axis. Find the coordinates of the unique point $P$ on $\overline{XY}$ such that none of these line segments (except $\overline{XY}$ itself) pass through $P$.


PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
To find the number of digits of \(2^{1998}\) and \(5^{1998}\), we use the formula for the number of digits of a number \(n\), which is given by \(\lfloor \log_{10} n \rfloor + 1\).

1. **Number of digits of \(2^{1998}\):**
   \[
   a = \lfloor \log_{10} (2^{1998}) \rfloor + 1 = \lfloor 1998 \log_{10} 2 \rfloor + 1
   \]
   Using the approximation \(\log_{10} 2 \approx 0.3010\):
   \[
   a = \lfloor 1998 \times 0.3010 \rfloor + 1 = \lfloor 601.398 \rfloor + 1 = 601 + 1 = 602
   \]

2. **Number of digits of \(5^{1998}\):**
   \[
   b = \lfloor \log_{10} (5^{1998}) \rfloor + 1 = \lfloor 1998 \log_{10} 5 \rfloor + 1
   \]
   Using the approximation \(\log_{10} 5 \approx 0.6990\):
   \[
   b = \lfloor 1998 \times 0.6990 \rfloor + 1 = \lfloor 1396.602 \rfloor + 1 = 1396 + 1 = 1397
   \]

3. **Sum of the number of digits:**
   \[
   a + b = 602 + 1397 = 1999
   \]

The final answer is \(\boxed{1999}\).