# Tuples with same coordinate sum

Some $$4$$-tuples of positive real numbers $$(a_1,b_1,c_1,d_1),\dots,(a_n,b_n,c_n,d_n)$$ are given, with $$\sum_{i=1}^na_i=\sum_{i=1}^nb_i=\sum_{i=1}^nc_i=\sum_{i=1}^nd_i=3.$$ It is known that there exists a partition of $$N=\{1,\dots,n\}$$ into three sets $$A_1,A_2,A_3$$ such that $$\sum_{A_1}a_i=\sum_{A_2}a_i=\sum_{A_3}a_i=1.$$ Analogous statements hold for $$b,c,d$$. Is it always possible to partition $$N$$ into two sets $$X,Y$$ so that $$\sum_X a_i,\sum_X b_i,\sum_Y c_i,\sum_Y d_i\geq 1?$$ When the last line has one $$X$$ and three $$Y$$'s, the answer is positive.