Find the largest constant $K>0$ such that for any $0\le k\le K$ and non-negative reals $a,b,c$ satisfying $a^2+b^2+c^2+kabc=k+3$ we have $a+b+c\le 3$.

(Dan Schwarz)
1. **Define VIP and Interesting Constants:**
   We define a constant \( k \) as "VIP" if for all nonnegative real numbers \( a, b, c \) satisfying \( a^2 + b^2 + c^2 + kabc = k + 3 \), we have \( a + b + c \leq 3 \).

   We define a constant \( k \) as "interesting" if for all nonnegative real numbers \( a, b, c \) satisfying \( a + b + c = 3 \), we have \( a^2 + b^2 + c^2 + kabc \geq k + 3 \).

2. **Prove Equivalence of VIP and Interesting Constants:**
   We need to show that a constant \( k \) is VIP if and only if it is interesting.

3. **VIP Implies Interesting:**
   Suppose \( k \) is VIP. For any nonnegative reals \( a, b, c \) such that \( a + b + c = 3 \), we need to show \( a^2 + b^2 + c^2 + kabc \geq k + 3 \).

   - Given \( a + b + c = 3 \), let \( c' \) be such that \( a + b + c' = 3 \).
   - Since \( k \) is VIP, for \( a, b, c' \) satisfying \( a^2 + b^2 + c'^2 + kabc' = k + 3 \), we have \( a + b + c' \leq 3 \).
   - Therefore, \( a^2 + b^2 + c'^2 + kabc' \geq k + 3 \).

4. **Interesting Implies VIP:**
   Suppose \( k \) is interesting. For any nonnegative reals \( a, b, c \) such that \( a^2 + b^2 + c^2 + kabc = k + 3 \), we need to show \( a + b + c \leq 3 \).

   - Given \( a^2 + b^2 + c^2 + kabc = k + 3 \), let \( c' \) be such that \( a + b + c' = 3 \).
   - Since \( k \) is interesting, for \( a, b, c' \) satisfying \( a + b + c' = 3 \), we have \( a^2 + b^2 + c'^2 + kabc' \geq k + 3 \).
   - Therefore, \( a + b + c' \leq 3 \).

5. **Determine the Largest \( K \):**
   To find the largest \( K \), we need to test the boundary conditions. Consider the case when \( a = b = c = 1 \):
   - \( a + b + c = 3 \)
   - \( a^2 + b^2 + c^2 + kabc = 1^2 + 1^2 + 1^2 + k \cdot 1 \cdot 1 \cdot 1 = 3 + k \)
   - For \( k \) to be interesting, \( 3 + k \geq k + 3 \), which is always true.

   Therefore, the largest \( K \) is determined by the boundary condition where \( a = b = c = 1 \).

The final answer is \( \boxed{ K = 1 } \)