Consider four sequences of numbers, $0 \le a_i, b_i, c_i, d_i \le 1$, suppose they satisfy the following constraints:

(1). $\sum_{i=1}^K a_i \ge 1/2 + \epsilon$;

(2). $\sum_{i=1}^K d_i \le 1/2 - \epsilon$;

(3). $a_i d_i = b_i c_i$ for all $i=1, \ldots, K$.

Is it true that \begin{equation} \sum_{i=1}^K (|a_i - b_i| + |a_i - c_i|) = \Omega(\epsilon) ? \end{equation}

Added remark after seeing the answer: Initially my Condition (1) was, (1). $\sum_{i=1}^K a_i, \sum_{i=1}^K b_i, \sum_{i=1}^K c_i \ge 1/2 + \epsilon$. But it turns out (see the answer below) that the inequality is true even without the requirement on b_i and c_i.