Finding contradiction in equations Question:- In the figure shown, Coefficient of friction between the blocks C and B is $0.4$. There is no friction between block C and ground.The system of blocks is released from rest in the shown situtation.Find the accelerations of masses.(Given $m_{B}=5kg , m_{C}=10kg , m_{A}=3kg$ )

I made two cases
$1)$ Block B and C move together.
$2)$ Block B and C will not move together.
When I solve equations for Case $1$, I find that block will move together
$$30-T=3a_{3}$$
$$T-f=5a_{5}$$
$$f=10a_{10}$$
$$f \leq 20$$
Note that if blocks B and C will move together then $a_{3}=a_{5}=a_{10}=a$
On solving we get $a=\frac {5}{3}m/s^2$ and $f=16.67N$
I got the following equations for case $2$.These equations should somehow contradict each other because case $1$ is correct.Despite my efforts I was unable to find how they will contradict.
\begin{align}
30-T&=3a_{3}\\
T-f&=5a_{5}\\
f&=10a_{10}\\
f&\leq {20}\\
a_{3}&>a_{10}\\
\end{align}
Note that $a_{3}=a_{5}$
Adding equation $1$ and $2$, we get
$$30-f=8a_{3}$$
So, $30-8a_{3}\leq 20$ and $10a_{10}\leq 20$
This gives $a_{3}\geq \frac54$ and $a_{10}\leq 2$
As $a_{3}>a_{10}$ but above inequalities of $a_{3}$ and $a_{10}$ satisfies it for certain  interval.
Can anybody help to find contradiction.
I have also asked same question on maths stackexchange since it involves inequalities
 A: There should be different between static frition coefficient $\mu_s$ and the kinetic friction ceofficient $\mu_k$, and $\mu_s$ must be greater than $\mu_k$. If $\mu_s \lt \mu_k$, there will induce non-physical phenomena.
Another concept is that the kenetic friction (as long as two badies have relative motion) is a constant $f_k = \mu_k N$ within the simple kenetic friction model. It is not less or equal than condition. Only in the static firction (when two bodies have no relative motion) $f_s \le \mu_s N$.
Therfore, in your problem, in setting $\mu_k = 0.4$, meaning $\mu_s \gt 0.4$.
In your case 1, the friction between Block B and Bolck C are static friction (there is no relative motion), $f=17 N$, $f_k = \mu_k 10 M_B=20 N$, these values satisfy $f \lt  \mu_k 10 M_B \le \mu_s 10 M_B$. It is ok to apply static friction for this case.
But in the case 2, there is relatice motion, the kinetic friction between Block B and Block C:
$$
  f_k = \mu_k 10 M_B = 20 N
$$
Once there is relative motion between B & C, the kenetic friction is a fixed number $20 N$.
The acceleration for Block C:
$$
  a_C = 20 / 10  = 2 m/s^2.
$$
and the acceleration for Block A and Block B system:
$$
   a_B = a_A = (30 - 20) / (3 + 5) = 1.25 m/s^2
$$
This results in block C an acceleration $2 m/s^2$, while the the system A & B has an acceleration $1.25 m/s^2$. It is not Physical.
Since the kenetic friction force is a fixed number as long as B and C having relative motion. There will be no flexible interval for the OP's condition to hold true.
