Problem 1148

Induction. For $n = 1$, the inequality to be proven holds (with an equality sign; since $a_1 + b_1 = c_1 + d_1$, in accordance with the assumption).
Let us fix a natural number $n \geq 1$ and assume the validity of the given theorem for this very number $n$. We need to prove its validity for $n + 1$.
Let then the numbers $a_i$, $b_i$, $c_i$, $d_i$ ($i = 1, \ldots, n + 1$) satisfy the conditions $0 \leq c_i \leq a_i \leq b_i \leq d_i$ and $a_i + b_i = c_i + d_i$. Denote

From the induction hypothesis, we have the inequality $A + B \leq C + D$, that is,

we need to prove that

By the conditions satisfied by the numbers $a_i$, $b_i$, $c_i$, $d_i$, the following equalities and inequalities hold:

where the expressions on both sides of each of the relations (1), (3), (4), (5) are non-negative. Inequalities directed consistently, binding non-negative numbers, can be multiplied side by side. Therefore, we multiply (1) by (4) and (3) by (5):

We add the obtained inequalities:

We have obtained the inductive thesis (2).
By the principle of induction, the theorem holds for any natural number $n$.