Since the median of the three integers is 9, we know that one of the integers is 9. Let's call the other two integers x and y.
Since the mean of the three integers is 9, we know that (x + y + 9)/3 = 9. This equation simplifies to x + y = 0.
Since the range of the three integers is 10, we know that the difference between the largest and smallest integers is 10. The largest integer is 9 + (10/2) = 14 and the smallest integer is 9 - (10/2) = 4.
We can now set up the following system of equations to find the values of x and y:
x + y = 0
4 <= x, y <= 14
The only solution to this system of equations is x = 7 and y = -7. Therefore, the three integers are 4, 7, and 9.