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!
