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

We are given the curve $y = x^3 - 6x^2 + 9x$ and the line $y = a$. We are asked: For how many integer values of $a$ do the curve and the line intersect at three distinct points?

Let's break down the steps to solve this:

Step-by-Step Plan

1. Set the Equations Equal:
   To find the intersection points, set $x^3 - 6x^2 + 9x = a$.

2. Form a Cubic Equation:
   Rearrange to $x^3 - 6x^2 + 9x - a = 0$.
   The number of real solutions to this equation tells us how many intersection points there are.

3. Find When There Are Three Distinct Real Roots:
   We need to determine for which integer values of $a$ this cubic equation has three distinct real roots.

4. Analyze the Cubic Function:

   • Find the critical points (where the slope is zero) by taking the derivative and setting it to zero.
   • Use these points to find the local maximum and minimum values of the cubic function.
   • The line $y = a$ will intersect the cubic at three points if $a$ is strictly between the local maximum and minimum values.

5. Count the Integer Values:

   • Find the integer values of $a$ that are strictly between the maximum and minimum values.

Now, let's execute each step in detail.

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Step 1: Set the Equations Equal

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

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Step 2: Find the Critical Points

Take the derivative of $f(x) = x^3 - 6x^2 + 9x$:

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

Set $f'(x) = 0$ to find the critical points.

Let's solve $3x^2 - 12x + 9 = 0$ for $x$. The critical points are $x = 1$ and $x = 3$.

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Step 3: Find the Local Maximum and Minimum Values

Let's substitute $x = 1$ and $x = 3$ into the original cubic function to find the local maximum and minimum values.

• For $x = 1$: $f(1) = (1)^3 - 6(1)^2 + 9(1)$
• For $x = 3$: $f(3) = (3)^3 - 6(3)^2 + 9(3)$

Let's calculate these values. At $x = 1$, the value is $f(1) = 4$.
At $x = 3$, the value is $f(3) = 0$.

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Step 4: Determine the Range for $a$

The cubic function will intersect the line $y = a$ at three distinct points if $a$ is strictly between the local maximum and minimum values.
Since $f(1) = 4$ and $f(3) = 0$, the possible values for $a$ are those strictly between 0 and 4.

So, $0 < a < 4$.

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Step 5: Count the Integer Values

The integer values of $a$ that satisfy $0 < a < 4$ are $1, 2, 3$.

So, there are 3 integer values.

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Final Answer:
Amanita, the number of integer values of $a$ for which the curve and the line intersect at three distinct points is $\boxed{3}$.