[b]p1.[/b] It’s currently $6:00$ on a $12$ hour clock. What time will be shown on the clock $100$ hours from now? Express your answer in the form hh : mm.


[b]p2.[/b] A tub originally contains $10$ gallons of water. Alex adds some water, increasing the amount of water by 20%. Barbara, unhappy with Alex’s decision, decides to remove $20\%$ of the water currently in the tub. How much water, in gallons, is left in the tub? Express your answer as an exact decimal.


[b]p3.[/b] There are $2000$ math students and $4000$ CS students at Berkeley. If $5580$ students are either math students or CS students, then how many of them are studying both math and CS?


[b]p4.[/b] Determine the smallest integer $x$ greater than $1$ such that $x^2$ is one more than a multiple of $7$.


[b]p5.[/b] Find two positive integers $x, y$ greater than $1$ whose product equals the following sum: 
$$9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 + 27 + 29.$$
Express your answer as an ordered pair $(x, y)$ with $x \le y$.


[b]p6.[/b] The average walking speed of a cow is $5$ meters per hour. If it takes the cow an entire day to walk around the edges of a perfect square, then determine the area (in square meters) of this square.


[b]p7.[/b] Consider the cube below. If the length of the diagonal $AB$ is $3\sqrt3$, determine the volume of the cube.
[img]https://cdn.artofproblemsolving.com/attachments/4/d/3a6fdf587c12f2e4637a029f38444914e161ac.png[/img]

[b]p8.[/b] I have $18$ socks in my drawer, $6$ colored red, $8$ colored blue and $4$ colored green. If I close my eyes and grab a bunch of socks, how many socks must I grab to guarantee there will be two pairs of matching socks?


[b]p9.[/b] Define the operation $a @ b$ to be $3 + ab + a + 2b$. There exists a number $x$ such that $x @  b = 1$ for all $b$. Find $x$.


[b]p10.[/b] Compute the units digit of $2017^{(2017^2)}$.


[b]p11.[/b] The distinct rational numbers $-\sqrt{-x}$, $x$, and $-x$ form an arithmetic sequence in that order. Determine the value of $x$.


[b]p12.[/b] Let $y = x^2 + bx + c$ be a quadratic function that has only one root. If $b$ is positive, find $\frac{b+2}{\sqrt{c}+1}$.


[b]p13.[/b] Alice, Bob, and four other people sit themselves around a circular table. What is the probability that Alice does not sit to the left or right of Bob?


[b]p14.[/b] Let $f(x) = |x - 8|$. Let $p$ be the sum of all the values of $x$ such that $f(f(f(x))) = 2$ and $q$ be the minimum solution to $f(f(f(x))) = 2$. Compute $p \cdot q$.


[b]p15.[/b] Determine the total number of rectangles ($1 \times 1$, $1 \times 2$, $2 \times 2$, etc.) formed by the lines in the figure below:
$  \begin{tabular}{ | l | c | c | r| }
    \hline
     &  &  & \\ \hline
     &  &  & \\ \hline
        &  &  & \\ \hline
  &  &  & \\
    \hline
  \end{tabular}
$


[b]p16.[/b] Take a square $ABCD$ of side length $1$, and let $P$ be the midpoint of $AB$. Fold the square so that point $D$ touches $P$, and let the intersection of the bottom edge $DC$ with the right edge be $Q$. What is $BQ$?
[img]https://cdn.artofproblemsolving.com/attachments/1/1/aeed2c501e34a40a8a786f6bb60922b614a36d.png[/img]

[b]p17.[/b] Let $A$, $B$, and $k$ be integers, where $k$ is positive and the greatest common divisor of $A$, $B$, and $k$ is $1$. Define $x\# y$ by the formula $x\# y = \frac{Ax+By}{kxy}$ . If $8\# 4 = \frac12$ and $3\# 1 = \frac{13}{6}$ , determine the sum $A + B + k$.


[b]p18.[/b] There are $20$ indistinguishable balls to be placed into bins $A$, $B$, $C$, $D$, and $E$. Each bin must have at least $2$ balls inside of it. How many ways can the balls be placed into the bins, if each ball must be placed in a bin?


[b]p19.[/b] Let $T_i$ be a sequence of equilateral triangles such that
(a) $T_1$ is an equilateral triangle with side length 1.
(b) $T_{i+1}$ is inscribed in the circle inscribed in triangle $T_i$ for $i \ge 1$.
Find $$\sum^{\infty}_{i=1} Area (T_i).$$


[b]p20.[/b] A [i]gorgeous [/i] sequence is a sequence of $1$’s and $0$’s such that there are no consecutive $1$’s. For instance, the set of all gorgeous sequences of length $3$ is $\{[1, 0, 0]$,$ [1, 0, 1]$, $[0, 1, 0]$, $[0, 0, 1]$, $[0, 0, 0]\}$. Determine the number of gorgeous sequences of length $7$.


PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
To solve the problem, we need to find the smallest integer \( x \) greater than 1 such that \( x^2 \) is one more than a multiple of 7. Mathematically, we need to find \( x \) such that:

\[ x^2 \equiv 1 \pmod{7} \]

We will check the squares of integers modulo 7 starting from \( x = 2 \).

1. Calculate \( 2^2 \mod 7 \):
   \[
   2^2 = 4 \quad \Rightarrow \quad 4 \mod 7 = 4
   \]
   Since \( 4 \neq 1 \), \( x = 2 \) is not a solution.

2. Calculate \( 3^2 \mod 7 \):
   \[
   3^2 = 9 \quad \Rightarrow \quad 9 \mod 7 = 2
   \]
   Since \( 2 \neq 1 \), \( x = 3 \) is not a solution.

3. Calculate \( 4^2 \mod 7 \):
   \[
   4^2 = 16 \quad \Rightarrow \quad 16 \mod 7 = 2
   \]
   Since \( 2 \neq 1 \), \( x = 4 \) is not a solution.

4. Calculate \( 5^2 \mod 7 \):
   \[
   5^2 = 25 \quad \Rightarrow \quad 25 \mod 7 = 4
   \]
   Since \( 4 \neq 1 \), \( x = 5 \) is not a solution.

5. Calculate \( 6^2 \mod 7 \):
   \[
   6^2 = 36 \quad \Rightarrow \quad 36 \mod 7 = 1
   \]
   Since \( 1 = 1 \), \( x = 6 \) is a solution.

Thus, the smallest integer \( x \) greater than 1 such that \( x^2 \) is one more than a multiple of 7 is \( x = 6 \).

The final answer is \( \boxed{6} \).