Do there exist polynomials $p(x), q(x), r(x)$ whose coefficients are positive integers such that $p(x) = (x^{2}-3x+3) q(x)$ and $q(x) = (\frac{x^{2}}{20}-\frac{x}{15}+\frac{1}{12}) r(x)$?
To determine if there exist polynomials \( p(x), q(x), r(x) \) with positive integer coefficients such that \( p(x) = (x^2 - 3x + 3) q(x) \) and \( q(x) = \left(\frac{x^2}{20} - \frac{x}{15} + \frac{1}{12}\right) r(x) \), we need to analyze the given conditions step by step.

1. **Express \( q(x) \) in terms of \( r(x) \):**
   \[
   q(x) = \left(\frac{x^2}{20} - \frac{x}{15} + \frac{1}{12}\right) r(x)
   \]
   To ensure \( q(x) \) has positive integer coefficients, \( r(x) \) must be chosen such that the product results in a polynomial with positive integer coefficients.

2. **Express \( p(x) \) in terms of \( q(x) \):**
   \[
   p(x) = (x^2 - 3x + 3) q(x)
   \]
   Similarly, \( q(x) \) must be chosen such that the product results in a polynomial with positive integer coefficients.

3. **Choose \( r(x) \) appropriately:**
   Let \( r(x) = 60(3x^2 + 4x + 5)(x^2 + 3x + 3) r_1(x) \), where \( r_1(x) \) has positive integer coefficients. This choice is made to ensure that the coefficients of \( q(x) \) and \( p(x) \) are positive integers.

4. **Calculate \( q(x) \):**
   \[
   q(x) = \left(\frac{x^2}{20} - \frac{x}{15} + \frac{1}{12}\right) 60(3x^2 + 4x + 5)(x^2 + 3x + 3) r_1(x)
   \]
   Simplify the expression inside the parentheses:
   \[
   \left(\frac{x^2}{20} - \frac{x}{15} + \frac{1}{12}\right) 60 = 3x^2 - 4x + 5
   \]
   Therefore,
   \[
   q(x) = (3x^2 - 4x + 5)(3x^2 + 4x + 5)(x^2 + 3x + 3) r_1(x)
   \]

5. **Calculate \( p(x) \):**
   \[
   p(x) = (x^2 - 3x + 3) q(x)
   \]
   Substitute \( q(x) \):
   \[
   p(x) = (x^2 - 3x + 3)(3x^2 - 4x + 5)(3x^2 + 4x + 5)(x^2 + 3x + 3) r_1(x)
   \]

6. **Ensure all coefficients are positive integers:**
   Since \( r_1(x) \) has positive integer coefficients, the product of polynomials with positive integer coefficients will also have positive integer coefficients. Therefore, \( p(x), q(x), \) and \( r(x) \) will all have positive integer coefficients.

Conclusion:
\[
\boxed{p(x) = (x^2 - 3x + 3)(3x^2 - 4x + 5)(3x^2 + 4x + 5)(x^2 + 3x + 3) r_1(x)}
\]
\[
\boxed{q(x) = (3x^2 - 4x + 5)(3x^2 + 4x + 5)(x^2 + 3x + 3) r_1(x)}
\]
\[
\boxed{r(x) = 60(3x^2 + 4x + 5)(x^2 + 3x + 3) r_1(x)}
\]