[b]p1.[/b] Let $A\% B = BA - B - A + 1$. How many digits are in the number $1\%(3\%(3\%7))$ ?


[b]p2. [/b]Three circles, of radii $1, 2$, and $3$ are all externally tangent to each other. A fourth circle is drawn which passes through the centers of those three circles. What is the radius of this larger circle?


[b]p3.[/b] Express $\frac13$ in base $2$ as a binary number. (Which, similar to how demical numbers have a decimal point, has a “binary point”.)


[b]p4. [/b] Isosceles trapezoid $ABCD$ with $AB$ parallel to $CD$ is constructed such that $DB = DC$. If $AD = 20$, $AB = 14$, and $P$ is the point on $AD$ such that $BP + CP$ is minimized, what is $AP/DP$?


[b]p5.[/b] Let $f(x) = \frac{5x-6}{x-2}$ . Define an infinite sequence of numbers $a_0, a_1, a_2,....$ such that $a_{i+1} = f(a_i)$ and $a_i$ is always an integer. What are all the possible values for $a_{2014}$ ?


[b]p6.[/b] $MATH$ and $TEAM$ are two parallelograms. If the lengths of $MH$ and $AE$ are $13$ and $15$, and distance from $AM$ to $T$ is $12$, find the perimeter of $AMHE$.


[b]p7.[/b] How many integers less than $1000$ are there such that $n^n + n$ is divisible by $5$ ?


[b]p8.[/b] $10$ coins with probabilities of $1, 1/2, 1/3 ,..., 1/10$ of coming up heads are flipped. What is the probability that an odd number of them come up heads?


[b]p9.[/b] An infinite number of coins with probabilities of $1/4, 1/9, 1/16, ...$ of coming up heads are all flipped. What is the probability that exactly $ 1$ of them comes up heads?


[b]p10.[/b] Quadrilateral $ABCD$ has side lengths $AB = 10$, $BC = 11$, and $CD = 13$. Circles $O_1$ and $O_2$ are inscribed in triangles $ABD$ and $BDC$. If they are both tangent to $BD$ at the same point $E$, what is the length of $DA$ ?


PS. You had better use hide for answers.
### Problem 1
Given the operation \( A \% B = BA - B - A + 1 \), we need to find the number of digits in the number \( 1 \% (3 \% (3 \% 7)) \).

1. First, compute \( 3 \% 7 \):
   \[
   3 \% 7 = 7 \cdot 3 - 7 - 3 + 1 = 21 - 7 - 3 + 1 = 12
   \]

2. Next, compute \( 3 \% 12 \):
   \[
   3 \% 12 = 12 \cdot 3 - 12 - 3 + 1 = 36 - 12 - 3 + 1 = 22
   \]

3. Finally, compute \( 1 \% 22 \):
   \[
   1 \% 22 = 22 \cdot 1 - 22 - 1 + 1 = 22 - 22 - 1 + 1 = 0
   \]

The number \( 0 \) has 1 digit.

The final answer is \( \boxed{ 0.0101010101\ldots_2 } \).