6. A triple of integers $(a,b,c)$ is said to be $\gamma$[i]-special[/i] if $a\le \gamma(b+c)$, $b\le \gamma(c+a)$ and $c\le\gamma(a+b)$. For each integer triple $(a,b,c)$ such that $1\le a,b,c \le 20$, Kodvick writes down the smallest value of $\gamma$ such that $(a,b,c)$ is $\gamma$-special. How many distinct values does he write down?

[i]Proposed by winnertakeover[/i]
1. **Understanding the problem**: We need to find the number of distinct values of \(\gamma\) such that the triple \((a, b, c)\) is \(\gamma\)-special. This means that for each integer triple \((a, b, c)\) with \(1 \leq a, b, c \leq 20\), we need to determine the smallest \(\gamma\) such that:
   \[
   a \leq \gamma(b + c), \quad b \leq \gamma(c + a), \quad c \leq \gamma(a + b)
   \]

2. **Simplifying the conditions**: Without loss of generality, assume \(a \geq b \geq c\). Then, the condition \(a \leq \gamma(b + c)\) will be the most restrictive. Therefore, the smallest \(\gamma\) can be expressed as:
   \[
   \gamma = \frac{a}{b + c}
   \]

3. **Range of \(b + c\)**: Since \(1 \leq b, c \leq 20\), the sum \(b + c\) ranges from 2 to 40. However, since \(a \geq b \geq c\), we need to consider the range of \(b + c\) relative to \(a\). Specifically, \(2 \leq b + c \leq 2a\).

4. **Distinct values of \(\gamma\)**: We need to find the number of distinct values of \(\frac{a}{b + c}\) for \(1 \leq a \leq 20\) and \(2 \leq b + c \leq 2a\). Notice that \(\frac{a}{b + c}\) must be in its simplest form (i.e., a reduced fraction).

5. **Counting distinct values**: For each \(a\), we need to count the number of distinct reduced fractions \(\frac{a}{b + c}\) where \(2 \leq b + c \leq 2a\). This can be done using Euler's totient function \(\phi\), which counts the number of integers up to \(n\) that are coprime with \(n\).

6. **Summing up**: For each \(a\), the number of distinct values of \(\frac{a}{b + c}\) is given by:
   \[
   \phi(a) - 1 \quad \text{for} \quad 2 \leq b + c \leq a
   \]
   \[
   \phi(a) \quad \text{for} \quad a < b + c \leq 2a
   \]
   Therefore, the total number of distinct values for each \(a\) is:
   \[
   (\phi(a) - 1) + \phi(a) = 2\phi(a) - 1
   \]

7. **Summing over all \(a\)**: We need to sum this expression for \(a\) from 1 to 20:
   \[
   \sum_{a=1}^{20} (2\phi(a) - 1)
   \]

8. **Adding the undercounted value**: We need to add 1 because we undercounted for \(a = 1\) (where \(\gamma = 1, \frac{1}{2}\)).

9. **Final calculation**: The final number of distinct values is:
   \[
   \sum_{a=1}^{20} (2\phi(a) - 1) + 1
   \]

The final answer is \( \boxed{ \sum_{a=1}^{20} (2\phi(a) - 1) + 1 } \)