Question

Question #43eb0

Answer 1

Warning: this is only a partial solution!

If `(5-sqrt(x))^2 = y - 20sqrt(2)`
then
`y = x-10sqrt(x)+25+20sqrt(2)`

`= (x+25) - 10(sqrt(x)+2sqrt(2))`

If `x` is an integer (given)
then for `y` to be an integer
`sqrt(x)-2sqrt(2)` must be an integer.

There is, of course, the obvious solution:
`x=8` since `sqrt(8) = 2sqrt(2)`
(which implies `y=33`).

The problem is in demonstrating that this is the only solution (or finding more solutions).

Anyone have further thoughts on this?