BALTIC-WAY 2018 Problem 2

A $100 \times 100$ table is given. For each $k, 1 \leq k \leq 100$, the $k$-th row of the table contains the numbers $1,2, \ldots, k$ in increasing order (from left to right) but not necessarily in consecutive cells; the remaining $100-k$ cells are filled with zeroes. Prove that there exist two columns such that the sum of the numbers in one of the columns is at least 19 times as large as the sum of the numbers in the other column.