Q.Two blocks of masses $2m$ (on the left) and $m$ (on the right) are placed in contact with each other on a smooth horizontal surface. In the first case, only a force $F_1$ is applied on the $2m$ block from the left. Later, only a force $F_2$ is applied on the $m$ block from the right. If the contact force acting at the interface of the two blocks is the same in both cases, then $F_1:F_2$ is (Physics Olympiad 2016)

When two blocks are pushed directly against each other (in physical contact, not connected by a string) on a frictionless horizontal surface, and a single external force $F$ is applied to the outer face of one of them, both blocks necessarily move together with one common acceleration, $a=\frac{F}{m_1+m_2}$ (found by treating the two blocks as a single combined system of total mass $m_1+m_2$).

To find the contact force — the force each block exerts on the other purely through direct pushing contact — isolate just the block that is not directly touched by $F$: the only horizontal force on that block is the contact push from its neighbour, so by Newton's second law that contact force equals (that block's own mass) $\times\,a$. This means the contact force depends specifically on the mass of the far block (the one away from where $F$ is applied), since the near block must "push through" the point of contact to accelerate the far one — By Newton's third law, the far block pushes back on the near block with an equal and opposite reaction. …