Amanita, let's focus on the problem inside the green rectangle, which is problem 12. Here’s what the problem says:

Find the number of integer values of $a$ such that the line $y = a$ intersects the curve $y = x^3 - 6x^2 + 9x$ at exactly three distinct points.

Let's break down the steps to solve this:

Step-by-Step Plan

1. Set Up the Intersection Equation:
   The intersection points of the curve and the line occur where $x^3 - 6x^2 + 9x = a$.

2. Rearrange the Equation:
   Rearranging gives $x^3 - 6x^2 + 9x - a = 0$.
   We need this cubic equation to have exactly three distinct real solutions for $x$.

3. Analyze the Cubic Function:
   The number of real roots of a cubic equation depends on the shape of its graph.
   We need to find the range of $a$ values for which the cubic has three distinct real roots.

4. Find the Local Maximum and Minimum:
   The cubic will have three real roots if the horizontal line $y = a$ passes between the local maximum and minimum of the cubic function.
   We need to find the $y$-values at these points.

5. Calculate the Values of $a$:
   The integer values of $a$ must be strictly between the local maximum and minimum values.

6. Count the Integer Values:
   Count how many integer values of $a$ satisfy this condition.

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Let's begin with Step 1 and Step 2:

We set the equations equal:

$x^3 - 6x^2 + 9x = a$

$x^3 - 6x^2 + 9x - a = 0$

Now, let's move to Step 3 and Step 4:
We need to find the local maximum and minimum of the cubic function $f(x) = x^3 - 6x^2 + 9x$.
To do this, we find the critical points by setting the derivative to zero.

The derivative is:

$f'(x) = 3x^2 - 12x + 9$

Let's solve $3x^2 - 12x + 9 = 0$ to find the critical points. Amanita, the critical points are at $x = 1$ and $x = 3$. Now, let's find the values of the cubic function at these points to determine the local maximum and minimum.

We substitute $x = 1$ and $x = 3$ into $f(x) = x^3 - 6x^2 + 9x$ to find the corresponding $y$-values. Amanita, at $x = 1$, the value of the function is $4$, and at $x = 3$, the value is $0$.

Now, let's determine which is the local maximum and which is the local minimum. For a cubic function like this, the smaller $x$ value (here, $x = 1$) will be the local maximum, and the larger $x$ value (here, $x = 3$) will be the local minimum.

So:

• Local maximum: $4$ at $x = 1$
• Local minimum: $0$ at $x = 3$

For the cubic to have three distinct real roots, the line $y = a$ must be between the local maximum and minimum, that is, $0 < a < 4$.

Now, let's count the integer values of $a$ that satisfy $0 < a < 4$. Amanita, let's list the integer values that satisfy $0 < a < 4$:

The integers between 0 and 4 are 1, 2, and 3.

So, there are 3 integer values of $a$ that make the line $y = a$ intersect the curve at exactly three distinct points.

Final Answer:
There are $\boxed{3}$ integer values of $a$ that satisfy the condition.