Consider a graph, where nodes represent pupils of this school and two nodes are connected by an edge if corresponding pupils are friends. Among all possible partitions of nodes on two sets $A$ and $B$ we choose a partition where $S = b \cdot S_A + a \cdot S_B$ is the smallest, where $S_A$ and $S_B$ denote the amount of edges inside groups $A$ and $B$. We show that this partition satisfies the required condition.

Indeed, let's assume that there exists a node $X$ with a degree at least $a+1$ in the group $A$, then all edges coming from $X$ add at least $b(a+1)$ to the sum $S$. Also, there are no more than $b$ nodes in the group $B$ that are adjacent to $X$. Thus if we shift $X$ to the group $B$, the sum $S$ will decrease by at least $b(a+1)$ and increase by $ba$. Therefore, $S$ decreases and we have a contradiction with our assumption of choosing the partition. Similarly, there is no node in the group $B$ with a degree $\ge b$.