Consider the set $M=\{1,2,3,...,2020\}.$ Find the smallest positive integer $k$ such that for any subset $A$ of $M$ with $k$ elements, there exist $3$ distinct numbers $a,b,c$ from $M$ such that $a+b, b+c$ and $c+a$ are all in $A.$
1. **Understanding the Problem:**
   We need to find the smallest positive integer \( k \) such that for any subset \( A \) of \( M = \{1, 2, 3, \ldots, 2020\} \) with \( k \) elements, there exist three distinct numbers \( a, b, c \) from \( M \) such that \( a+b \), \( b+c \), and \( c+a \) are all in \( A \).

2. **Initial Observations:**
   - The set \( M \) contains 2020 elements.
   - We need to ensure that for any subset \( A \) of \( M \) with \( k \) elements, the condition holds.

3. **Triangle Inequality and Even Sum:**
   - The problem hints at the triangle inequality and the sum being even.
   - For three numbers \( a, b, c \) to form a triangle, they must satisfy the triangle inequality: \( a + b > c \), \( b + c > a \), and \( c + a > b \).
   - Additionally, \( a + b + c \) must be even for \( a+b \), \( b+c \), and \( c+a \) to be integers in \( A \).

4. **Choosing \( k \):**
   - We need to find the smallest \( k \) such that any subset \( A \) of \( M \) with \( k \) elements contains three numbers \( a, b, c \) satisfying the conditions.
   - Consider the worst-case scenario where \( A \) is chosen to avoid such triples.

5. **Constructing a Subset:**
   - If we choose all odd numbers from \( M \), we get 1010 odd numbers.
   - If we choose all even numbers from \( M \), we get 1010 even numbers.
   - In both cases, we do not have three numbers \( a, b, c \) such that \( a+b \), \( b+c \), and \( c+a \) are all in \( A \).

6. **Increasing \( k \):**
   - If \( k \geq 1011 \), then any subset \( A \) of \( M \) with \( k \) elements must contain at least one odd and one even number.
   - This ensures that there are three numbers \( a, b, c \) such that \( a+b \), \( b+c \), and \( c+a \) are all in \( A \).

7. **Verification:**
   - Suppose \( |A| \geq 1012 \). Then, \( A \) must contain at least two consecutive pairs of numbers, say \( a < a+1 < b < b+1 \).
   - The triples \( (a, b, b+1) \) and \( (a+1, b, b+1) \) form triangles, and one of them will have an even sum, ensuring the required condition.

Conclusion:
The smallest positive integer \( k \) such that for any subset \( A \) of \( M \) with \( k \) elements, there exist three distinct numbers \( a, b, c \) from \( M \) such that \( a+b \), \( b+c \), and \( c+a \) are all in \( A \) is \( k = 1011 \).

The final answer is \( \boxed{1011} \)