It looks like you've uploaded an image containing a set of mathematical problems. Let me process the image and provide solutions to the problems.

I'll start by reading the image and identifying the questions.Here is a transcription of the questions from the image:

1. If $\frac{a+b+c}{abc} = \frac{3}{2}$, then prove that $a + b + c = \frac{3abc}{a+b+c}$ when each ratio equals $\frac{3}{2}$.

2. Two circles $C_1$ and $C_2$ intersect with their radii $r_1$ and $r_2$ such that $r_1 \neq r_2$ and $r_2 = 2r_1$. What quantity remains the same in both circles when the radii increase along with the distance between their centers?

3. Given that $x + y + z = 1$ and $x^2 + y^2 + z^2 = 1$, prove that $xy + yz + zx = 0$.

4. If $x^2 + y^2 + z^2 = 1$ and $x + y + z = 0$, then show that $xy + yz + zx = -\frac{1}{2}$.

5. Is it possible to fit an equilateral triangle inside a square so that each side of the triangle is parallel to one side of the square? Prove it.

6. A tennis coach gives an array $[3, 5, 9, 10, 14]$ to his trainee, who adds a double to each element of the array. The final array is $[6, 10, 18, 20, 28]$. The trainee claims that the original array must have had two numbers. Is the trainee correct? If not, determine the number of elements that must have been in the original array.

7. Three school students decide to raise money by selling lemonade. They sell lemonade at $2 per cup, and each cup has 300 ml of lemonade. After the first day, they count their earnings and realize they sold a total of 1200 ml of lemonade. Calculate the amount of money they earned.

8. Assume the function $f(x)$ is defined as $f(x) = \frac{2}{3}x + 4$. What is the value of $f(3)$?

9. The sequence $a_n$ is given by $a_n = a_{n-1} + 3$ with $a_1 = 2$. Find the value of $a_5$.

10. Explain the difference between a real number and a complex number. Then, explain how to add $2 + 3i$ and $1 + 4i$, where $i$ is the imaginary unit.

11. Explain how to find the roots of the quadratic equation $x^2 - 3x + 2 = 0$ using the quadratic formula.

12. If the remainder when $x^2 + 2x + 1$ is divided by $x + 1$ is 0, what does that tell you about the roots of the quadratic polynomial?

Would you like me to solve any specific problem from this list or provide details on any of them?

Here are five related questions that might help further your understanding:

1. How can you use the quadratic formula to solve any quadratic equation?
2. What properties distinguish an equilateral triangle from other triangles?
3. How can the discriminant of a quadratic equation determine the nature of its roots?
4. What is the relationship between the sum and product of the roots of a quadratic equation?
5. How does adding complex numbers differ from adding real numbers?

Tip: When solving equations involving multiple variables, it's often helpful to express one variable in terms of others to simplify the problem.