Integrate step-by-step reasoning and Python code to solve math problems using the following guidelines:

- Analyze the question; list some knowledge points related to the question and beneficial for problem solving.
- Write python functions to solve the problem; the function should not take any arguments.
- Present the final result in LaTeX using a `\boxed{}` without any units.
- Utilize the `pi` symbol and `Rational`` from Sympy for $\pi$ and fractions, and simplify all fractions and square roots without converting them to decimal values.

Here are some examples you may refer to:

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Question: Convert the point $(0, -3 \sqrt{3}, 3)$ in rectangular coordinates to spherical coordinates.  Enter your answer in the form $(\rho,\theta,\phi),$ where $\rho > 0,$ $0 \le \theta < 2 \pi,$ and $0 \le \phi \le \pi.$

Solution:
To convert the point $(x, y, z)$ in rectangular coordinates to spherical coordinates $(\rho, \theta, \phi)$, we use the following formulas:
1. $\rho = \sqrt{x^2 + y^2 + z^2}$
2. $\theta = \arctan{\frac{y}{x}}$
3. $\phi = \arccos{\frac{z}{\rho}}$

Let's write a function:

```python
from sympy import sqrt, atan2, acos, pi

def rectangular_to_spherical():
    x, y, z = 0, -3*sqrt(3), 3
    rho = sqrt(x**2 + y**2 + z**2)
    theta = atan2(y, x)
    phi = acos(z/rho)
    return rho, theta, phi

spherical_coordinates = rectangular_to_spherical()
print(spherical_coordinates)
```
```output
(6, -pi/2, pi/3)
```

The required form of the answer is $(\rho, \theta, \phi)$, where $\rho > 0$, $0 \le \theta < 2\pi$, and $0 \le \phi \le \pi$. Since our initial output has $\theta = -\pi/2$, which doesn't meet the condition for $\theta$, we add $2\pi$ to adjust its value.

Hence, the spherical coordinates for the point $(0, -3\sqrt{3}, 3)$ are $\boxed{(6, \frac{3\pi}{2}, \frac{\pi}{3})}$.

---

Question: Compute $1011_2 + 101_2 - 1100_2 + 1101_2$.  Express your answer in base 2.

Solution:
Each digit of a binary number can only be 0 or 1, and we can use int(the_binary_string, 2) to represent a binary integer in Python.
Let's write a single python function for binary computation:

```python
def binary_sum_diff():
    # Convert binary to decimal
    num1 = int("1011", 2)
    num2 = int("101", 2)
    num3 = int("1100", 2)
    num4 = int("1101", 2)

    # Perform the operations
    result = num1 + num2 - num3 + num4

    # Convert the result back to binary
    result_binary = format(result, "b")

    return result_binary

result = binary_sum_diff()
print(result)
```
```output
10001
```
The answer expressed in base 2 is $\boxed{10001_2}$.

---

Question: Find the matrix that corresponds to projecting onto the vector $\begin{pmatrix} 2 \\ -3 \end{pmatrix}.$

Solution:
The projection matrix onto a vector $v$ is given by $P = \frac{vv^T}{v^T v}$, we can calculate it and simplify the result using sympy. Let's write a function:
```python
from sympy import Matrix, simplify

def projection_matrix():
    v = Matrix([2, -3])

    # Calculate the projection matrix
    P = (v * v.T) / (v.T * v)[0]

    # Simplify the elements
    P = P.applyfunc(simplify)

    return P

result = projection_matrix()
print(result)
```
```output
Matrix([[4/13, -6/13], [-6/13, 9/13]])
```

The matrix is $\boxed{\begin{pmatrix} \frac{4}{13} & -\frac{6}{13} \ -\frac{6}{13} & \frac{9}{13} \end{pmatrix}}.$

---

Question: Solve the inequality
\[\frac{x^2 - 25}{x + 5} < 0.\]

Solution:
Let's use `sympy` to solve the inequality:

```python
from sympy import symbols, simplify
from sympy.solvers.inequalities import solve_univariate_inequality
from sympy.core.relational import LessThan

def solve_inequality():
    x = symbols('x')
    expression = (x**2 - 25) / (x + 5)
    inequality = LessThan(expression, 0)
    solution = solve_univariate_inequality(inequality, x, relational=False)
    simplified_solution = simplify(solution)
    
    return simplified_solution

result = solve_inequality()
print(result)
```
```output
Union(Interval.open(-oo, -5), Interval.Lopen(-5, 5))
```
The answer is $\boxed{(-\infty,-5)\cup(-5,5)}$

---

Question: In the figure, triangles $ABC$ and $BCD$ are equilateral triangles. What is the value of $AD \div BC$ when expressed in simplest radical form?

[asy]
draw((0,0)--(5,8.7)--(10,0)--cycle);
draw((10,0)--(15,8.7)--(5,8.7));
label("$A$",(0,0),SW);
label("$B$",(5,8.7),N);
label("$C$",(10,0),SE);
label("$D$",(15,8.7),NE);
[/asy]

Solution:
We found that the equilateral triangles ABC and BCD share a common side, BC. For this problem, we can assume that the sides of both triangles are of length 1, and then express the length of AD in terms of this side length.
```python
from sympy import Rational, sqrt, simplify

def ad_divided_by_bc():

    x = Rational(1, 1)  # Side length of equilateral triangles

    ad_squared = 2 * x**2 * (1 + Rational(1, 2))  # Using the law of cosines with cos(2*pi/3) = -1/2
    ad = sqrt(ad_squared)

    bc = x # BC is the side length of the equilateral triangles

    simplified_ratio = simplify(ad / bc)

    return simplified_ratio

result = ad_divided_by_bc()
print(result)
```
```output
sqrt(3)
```
The value of $AD \div BC$ is $\boxed{\sqrt{3}}$.

---
