Problem R-5 - solution

SOLUTION TO PROBLEM R-5

Let a₁, a₂, a₃, a₄, a₅ be the five numbers. Then

$\begin{displaymath}(a_1+a_2+a_5-a_3-a_4)(a_1+a_3+a_4-a_2-a_5)=a_1^2-(a_2+a_5-a_3-a_4)^2 \leq a_1^2 . \end{displaymath}$

Similarly it follows that

$\begin{displaymath}(a_2+a_3+a_1-a_4-a_5)(a_2+a_4+a_5-a_3-a_1)\leq a_2^2, \end{displaymath}$

$\begin{displaymath}(a_3+a_4+a_2-a_5-a_1)(a_3+a_5+a_1-a_4-a_2)\leq a_3^2, \end{displaymath}$

$\begin{displaymath}(a_4+a_5+a_3-a_1-a_2)(a_4+a_1+a_2-a_5-a_3)\leq a_4^2, \end{displaymath}$

$\begin{displaymath}(a_5+a_1+a_4-a_2-a_3)(a_5+a_2+a_3-a_1-a_4)\leq a_5^2. \end{displaymath}$

Multiply the inequalities to get the desired result.

Questions and/or comments should be directed to Judy Downey or Griff Elder

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Last modified: Thu Feb 14 20:19:51 CST 2002