Baltic Way shortlist

Observe that the sum of numbers in the first column is at most $1\cdot 100=100$, the sum in the first and second columns is at most $1\cdot 100+2\cdot 99$, the sum in the first, second and third columns is at most $1\cdot 100+2\cdot 99+3\cdot 98$, etc. But the sum of all nonzero numbers equals ${\sum }_{i=1}^{100}i(101-i)$, therefore the sum in the columns from $31$-th to $100$-th is at least $$\sum _{i=31}^{100}i(101-i)=\sum _{i=1}^{70}i(101-i)=101\sum _{i=1}^{70}i-\sum _{i=1}^{70}{i}^2=35\cdot 71(101-141/3)=70\cdot 27\cdot 71.$$ Therefore one of these columns has a sum at least $27\cdot 71=1917$. Therefore the ratio of sums in this column and in the first one is more than $19$.