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such that ![\[f(xy-1)+f(x)f(y)=2xy-1,\]](http://latex.artofproblemsolving.com/8/8/8/888ca39f2b7f8cec6d6426bee28d40eade40a66e.png)
for any real numbers 
and 
, consider the 
fractions 
The product of these fractions equals 
, but if you reciprocate (i.e. turn upside down) some of the fractions, the product will change. Can you make the product equal 1? Find all values of for which this is possible and prove that you have found them all.
be a function satisfying
for all integers 
and 
. Show that there exist positive integers 
and such that 
for all integers .
+1 w
is the midpoint of the base 
of trapezoid 
. On base 
, point 
is selected. Line 
intersects line 
at point 
, and the perpendicular from on the bases intersects line 
at point 
. Prove that 
.
is the midpoint of the base of trapezoid . On base , point is selected. Line intersects line at point , and the perpendicular from on the bases intersects line at point . Prove that .
cuts at 
. So by Thales's theorem,
. But 
, so 
. We also have that 
, hence 
is isoceles at . So 
and 
.
intersects 
.Enough to show: 
or 
.notice that 
and 
are homothetic and 
are collinear so