A Problem That Has a Chance of Becoming a Theorem // Problema, kuri gali tapti teorema

          I heard the expression in the title from a somewhat megalomaniac fellow mathematician.

          It does, and we will try to prove the following formula :

                    Let be the nonzero numbers  $a_1,\;\dots a_m$. The following equation holds :

$a_1\cdot \left ( \frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+\dots +\frac{1}{a_m}\right )+(a_2-a_1)\cdot \left ( \frac{1}{a_2}+\frac{1}{a_3}+\dots +\frac{1}{a_m}\right )+$

$+(a_3-a_2)\cdot \left (\frac{1}{a_3}+\dots +\frac{1}{a_m}\right )+\dots+(a_{m-1}-a_{m-2})\cdot \left ( \frac{1}{a_{m-1}}+\frac{1}{a_m}\right )+$

$+(a_m-a_{m-1})\cdot \frac{1}{a_m}=m \tag{1}$

I seem to see that a "proof without words" is imminent.