What are the possible values of these letters?

Out of all the questions I answered in a math reviewer, this one killed me (and 7 more).

Let $J, K, L, M, N$ be five distinct positive integers such that $$ \frac{1}{J} + \frac{1}{K} + \frac{1}{L} + \frac{1}{M} + \frac{1}{N} + \frac{1}{JKLMN} = 1. $$ Then, what is $J + K + L + M + N$?

I have been thinking about this for nearly 6 days.

Induction could lead you to the answer. The equation is :

$$ \frac 1 {x_1} + \frac 1 {x_2} + \dots + \frac 1 {x_{n}} + \frac 1 { x_1 x_2 \dots x_{n}} = 1 $$ Case $ n = 0 $: the empty set solves the equation as an empty product is 1

Case $ n = 1 $: the obvious solution is $ x_1 = 2 $.

Case $ n = 2 $: a bit more difficult, but you can find $ x_1 = 2, x_2 = 3 $. Doing this, I noticed one thing: assuming that you solved the $ (n-1) $-th equation, you can pick $ x_n $ so that $ + \frac 1 {x_{n}} $ in the first part of the equation compensates the factor $ \frac 1 {x_n} $. Let’s check.

Case any $ n $: assuming that $ x_1, \dots x_{n} $ solves the equation, we require $ x_{n+1} $ so that

$$ \frac 1 {x_n} + \frac 1 {x_2} + \dots + \frac 1 {x_{n+1}} + \frac 1 { x_1 x_2 \dots x_{n+1}} = \frac 1 {x_1} + \frac 1 {x_2} + \dots + \frac 1 {x_n} + \frac 1 { x_1 x_2 \dots x_n} $$

Removing identical summands:

$$ \frac 1 {x_{n+1}} + \frac 1 { x_1 x_2 \dots x_{n+1}} = \frac 1 { x_1 x_2 \dots x_n } $$ Multiplying tops by $ x_1 x_2 \dots x_{n+1} $ : $$ x_1 x_2 \dots x_{n} + 1 = x_{n+1} $$

Solved!

A start, on my phone.

Assume $j<k<l<m<n.$

Then j=2 or 3 because 1 makes the sum too large and 4 makes it too small.

Therefore the left without 1/j is 1/2 or 2/3. You can get a tree of possiblities by continuing in this way.

Another tack:

Clear fractions to get

$klmn+jlmn+jkmn+jkln+jklm+1=jklmn$

$klmn+j(...)+1=jklmn$

$j(klmn-...)=klmn+1$.

Therefore $j|(klmn+1)$ (and similarly for the others) so that j is relatively prime to the others.

Therefore all the variables are pairwise relatively prime.