Determine all integer solutions (a,b,c) with $c\leq 94$ for which:
$(a+\sqrt c)^2+(b+\sqrt c)^2 = 60 + 20\sqrt c$
1. Start with the given equation:
   \[
   (a + \sqrt{c})^2 + (b + \sqrt{c})^2 = 60 + 20\sqrt{c}
   \]

2. Expand both sides:
   \[
   (a + \sqrt{c})^2 + (b + \sqrt{c})^2 = a^2 + 2a\sqrt{c} + c + b^2 + 2b\sqrt{c} + c
   \]
   \[
   = a^2 + b^2 + 2c + 2(a + b)\sqrt{c}
   \]

3. Equate the expanded form to the right-hand side:
   \[
   a^2 + b^2 + 2c + 2(a + b)\sqrt{c} = 60 + 20\sqrt{c}
   \]

4. Separate the rational and irrational parts:
   \[
   a^2 + b^2 + 2c = 60
   \]
   \[
   2(a + b)\sqrt{c} = 20\sqrt{c}
   \]

5. From the irrational part, assuming \(\sqrt{c} \neq 0\):
   \[
   2(a + b) = 20 \implies a + b = 10
   \]

6. Substitute \(a + b = 10\) into the rational part:
   \[
   a^2 + b^2 + 2c = 60
   \]

7. Express \(b\) in terms of \(a\):
   \[
   b = 10 - a
   \]

8. Substitute \(b = 10 - a\) into the equation:
   \[
   a^2 + (10 - a)^2 + 2c = 60
   \]
   \[
   a^2 + 100 - 20a + a^2 + 2c = 60
   \]
   \[
   2a^2 - 20a + 100 + 2c = 60
   \]
   \[
   2a^2 - 20a + 2c = -40
   \]
   \[
   a^2 - 10a + c = -20
   \]

9. Solve for \(c\):
   \[
   c = 20 - a^2 + 10a
   \]

10. Since \(c\) must be an integer and \(c \leq 94\), we need to check values of \(a\) that satisfy this condition:
    \[
    20 - a^2 + 10a \leq 94
    \]
    \[
    -a^2 + 10a \leq 74
    \]
    \[
    a^2 - 10a + 74 \geq 0
    \]

11. Solve the quadratic inequality:
    \[
    a^2 - 10a + 74 \geq 0
    \]
    The discriminant of \(a^2 - 10a + 74\) is:
    \[
    \Delta = 10^2 - 4 \cdot 1 \cdot 74 = 100 - 296 = -196
    \]
    Since the discriminant is negative, the quadratic expression is always positive, so the inequality holds for all \(a\).

12. Check integer values of \(a\) from 0 to 10 (since \(a + b = 10\)):
    \[
    c = 20 - a^2 + 10a
    \]
    - For \(a = 0\), \(c = 20\)
    - For \(a = 1\), \(c = 29\)
    - For \(a = 2\), \(c = 36\)
    - For \(a = 3\), \(c = 41\)
    - For \(a = 4\), \(c = 44\)
    - For \(a = 5\), \(c = 45\)
    - For \(a = 6\), \(c = 44\)
    - For \(a = 7\), \(c = 41\)
    - For \(a = 8\), \(c = 36\)
    - For \(a = 9\), \(c = 29\)
    - For \(a = 10\), \(c = 20\)

13. Verify the solutions:
    - For \(a = 3\), \(b = 7\), \(c = 41\)
    - For \(a = 4\), \(b = 6\), \(c = 44\)
    - For \(a = 5\), \(b = 5\), \(c = 45\)
    - For \(a = 6\), \(b = 4\), \(c = 44\)
    - For \(a = 7\), \(b = 3\), \(c = 41\)

14. The valid triples \((a, b, c)\) are:
    \[
    (3, 7, 41), (4, 6, 44), (5, 5, 45), (6, 4, 44), (7, 3, 41)
    \]

The final answer is \( \boxed{ (3, 7, 41), (4, 6, 44), (5, 5, 45), (6, 4, 44), (7, 3, 41) } \)