Question

Where do I start with this?

`vwxyz=v+w+x+y+z`

Find the maximum possible value of `x`
where `v, w, x, y, z > 0`

Answer 1

Maximum for natural numbers is `x=5`

Explanation:

Given:

`vwxyz = v+w+x+y+z`

For solutions in natural numbers, we can look at possible combinations of small values.

Note that:

• If `v=w=x=y=z=1` then `vwxyz=1 < 5 =v+w+x+y+z`

• If `v=w=x=y=z=2` then `vwxyz=32 > 10 = v+w+x+y+z`

The minimum possible values of `v, w, y, z` are `v=w=y=z=1`, resulting in:

`x = x+4" "` which has no solutions.

Considering the case where exactly one of `v, w, y, z > 1`, and putting `v=2`, `w=y=z=1` we get:

`2x = x+5" "=>" "color(blue)(x = 5)`

Giving `v` larger values either results in no integer solution or smaller integer solutions for `x`.

Considering the case where exactly two of `v, w, y, z > 1`, and putting `v=w=2`, `y=z=1` we get:

`4x = x+6" "=>" "color(blue)(x = 2)`

Giving `v, w` larger values results in smaller values of `x` or no integer solution.

Considering the case where exactly three of `v, w, y > 1`, and putting `v=w=y=2`, `z=1` we get:

`8x = x+7" "=>" "color(blue)(x=1)`

Giving `v, w, y` larger values results in no integer solution for `x`.

Footnote

If `v, w, x, y, z > 0` are not restricted to integers, then simplify the problem by considering the case where `v=w=y=z` and hence:

`v^4x = x+4v" "=>" "x = (4v)/(v^4-1)`

Note that:

`lim_(v->1+) (4v)/(v^4-1) = +oo`

So `x` can be arbitrarily large.

Answer 2

Maximum value `x=5`

Explanation:

Another approach...

Given:

`vwxyz = v+w+x+y+z" "` with `v, w, x, y, z > 0`

Looking for solutions in positive integers.

Note that if `v=w=y=z=1` then:

`x = x+4" "` which has no solutions.

So at least one of `v, w, x, y, z > 1`.

Considering the given equation as expressing `x` as a function of `v`, with the other variables constant, we have:

`((wyz)v - 1)x = v+(w+y+z)`

So:

`x = (v+(w+y+z))/((wyz)v-1)`

`color(white)(x) = ((wyz)v-1+1+(wyz)(w+y+z))/((wyz)((wyz)v-1))`

`color(white)(x) = 1/(wyz)+(1+(wyz)(w+y+z))/((wyz)((wyz)v-1))`

So:

`(del x)/(del v) = -(1+(wyz)(w+y+z))/((wyz)v-1)^2`

Note that if `v, w, y, z >= 1` with at least one `> 1` then the denominator is positive and `(del x)/(del v) < 0`

So increasing the value of `v` will decrease the value of `x`.

Similarly for `w, y, z`.

Hence the maximum value for `x` is obtained with `v=2`, `w=y=z=1` or permutations thereof...

`2x = x+5" "=>" "color(blue)(x=5)`