67th Romanian Mathematical Olympiad

a) An example is given in figure 1.

1	2	3	10
8	7	6	5
9	4	11	12
16	15	14	13

Figure 1

b) The sum of the numbers written in the table is $1+2+\cdots +16=136$.

Since $136=4\cdot 34$, either the sum of the numbers in each column is $34$, or there exists a column with sum at least $35$. In the first case $S$ does not exist and in the second case $S$ is at least $35$.

1	2	3	4
8	7	6	5
9	10	11	12
16	15	13	14

Figure 2

An example with $S=35$ is given in figure 2, so the minimum possible value of $S$ is $35$.