by djmathman, Jul 24, 2015, 2:52 AM ![\[\int_0^1\int_0^1 xy(-1)^{\lfloor 1/x-1/y\rfloor}\,dx\,dy,\]](http://latex.artofproblemsolving.com/e/d/c/edc80c125218c9dc0ccfa46a0d366dbb8ced9269.png)
where 
denotes the floor function.
by
mssmath, Jul 25, 2015, 1:16 AM
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.
and 
have different parities for all 
,
whenever 
is not an integer. Thus almost everything cancels out and the region of integration collapses from the ![$[0,1]\times[0,1]$](http://latex.artofproblemsolving.com/a/6/a/a6aa56acfcb95ab5be4a1c4210c59219c425167b.png)
square to the union of disjoint one-dimensional curves corresponding to the points 
which satisfy 
for some integer 
.
.