I want to know if this problem is NP-hard? The problem: Given a non-negative integer-valued matrix of size $3\times n$ of the form $$\begin{bmatrix}a_1 & \ldots & a_n\\b_1 & \ldots & b_n\\a_1b_1 & \ldots & a_nb_n\\\end{bmatrix},$$where $a_i,b_i$ are non-negative integers. Given four non-negative integer numbers $b<n$, $c_1$, $c_2$ and $c_3$. The goal: Find a subset of the columns of cardinality at most $b$ such that (i) the sum of the first row over the subset of the columns is greater than $c_1$, (ii) the sum of the second row over the subset of the columns is greater than $c_2$ and (iii) the sum of the third row over the subset of the columns is less than $c_3$. Return no if no such subset exists. That is to say, is there exists a set $S$ of cardinality at most $b<n$ such that:$$\sum_{i\in S}a_i\geq c_1,\\\sum_{i\in S}b_i\geq c_2,\\\sum_{i\in S}a_ib_i\leq c_3,$$How to show that this problem is NP-hard?Attempt:Here is a similar NP-hard problem that I asked. This problem has as instance $(n,b,c,\mathbf{A})$ where $\mathbf{A}$ is given by $$\begin{bmatrix}x_1 & \ldots & x_n\\y_1 & \ldots & y_n\\\end{bmatrix},$$I tried to reduce the previous problem to the new one as follows: choose $c_1=c_2=c$ and choose $c_3$ very big and create the matrix as follows:$$\begin{bmatrix}x_1 & \ldots & x_n\\y_1 & \ldots & y_n\\x_1y_1 & \ldots & x_ny_n\\\end{bmatrix},$$
Select a subset of the columns in $3\times n$ matrix, is it NP-hard?
reductions;np hard