In maths olympiads, I frequently encounter situations where I have two compare two expressions $a_{1}b_{1}c_{1}...z_{1}$ and $a_{2}b_{2}c_{2}...z_{2}$ where $a_{1} + b_{1} + c_{1} + ... + z_{1} = a_{2} + b_{2} + c_{2} + ... +z_{2}$. We assume that $a_{1} < b_{1} < c_{1}...< z_{1}$ and $a_{2} < b_{2} < c_{2}...< z_{2}$ .

In the two variable case, it is quite simple:

$a_{1}b_{1} < a_{2}b_{2}$ if and only if $a_{1} < a_{2} < b_{2} < b_{1}$.

What about the three variable case ? When does $a_{1}b_{1}c_{1} < a_{2}b_{2}c_{2}$ hold?

Is this generalisable for the nth case? Is there a theorem that I can quote?

In this question I am only concerned with positive real numbers, but results about positive and negative numbers are welcome too.

  • 1
    $\begingroup$ 1) You'd want to stick to positive numbers because negatives numbers can muck up your inequality multiplication. 2) Check out smoothing, where you convert the expression term by term. IE Try something like $ a_1b_1c_1 < a_1(b_1+c_1 - c_2)c_2 < a_2b_2c_2$ under suitable conditions (find them). $\endgroup$
    – Calvin Lin
    Jan 6 at 13:28
  • $\begingroup$ @CalvinLin Smoothing is exactly the method I was looking for! In your method, the conditions are $c_{2} < c_{1}$ and $a_{1} < a_{2}$ right? $\endgroup$ Jan 6 at 15:22
  • $\begingroup$ The better way to write it is $ c_1 \leq c_2, b_1 + c_1 \leq b_2 + c_2$, because that allows you to easily extend to more variables. Note, of course, that this applies to a certain set of cases, and not necessarily all cases. $\endgroup$
    – Calvin Lin
    Jan 6 at 18:16


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