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 Just prove it (Posted on 2013-10-20)
Prove that:
If (x+sqrt{1+y^2})*(y+sqrt{1+x^2})=1,
then
x+y=0 .

 No Solution Yet Submitted by Ady TZIDON No Rating

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 Doing it backwards | Comment 1 of 4
Well, it is easy enough to prove that if
x+y=0
then
(x+sqrt{1+y^2})*(y+sqrt{1+x^2})=1

Just substitute y = -x in the messy formula, and do the multiplication, and simplify.

Unfortunately, that doesn't prove that this is the only solution.

Squaring both sides of the equation didn't seem to help.

I guess that some more clever substitution is called for, but I'm not finding it.  I tried c = x + y, with a hope of proving that c = 0, but substituting y = c-x didn't seem to help either.

This problem is a difficulty 2?  I will be very interested to see the solution.

 Posted by Steve Herman on 2013-10-21 09:19:44

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