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such that for all 
,![\[f(x+yf(x))+y = xy + f(x+y).\]](http://latex.artofproblemsolving.com/c/f/3/cf3d20a041c27244e90876119b4568b7a3e13c03.png)
be a positive integer, and let 
be the set 
.
and 
be difined as:
Which is larger, 
or 
?
be reals such that 
and 
Prove that
Where 
such that 
.
has at most one solution continuous for 
are complex numbers such that
and if 
denotes the sum of all products of these numbers taken 
at a time, prove that
whenever the denominator of the left-hand side is different from 
.
for any positive reals satifying 
?
are real number satisfying
show that the equation 
has at least one real root.
converge?
,where 
,and it is positiv integer number.
to find the third side 
He uses logarithms as follows. He finds 
and doubles it; adds to that the double of 
subtracts the sum of the logarithms of 
and 
divides the result by 
and takes the anti-logarithm. Although his method may be open to suspicion his computation is accurate. What are the necessary and sufficient conditions on the triangle that this method should yield the correct result?![\[ f(x) = \sum_{i=0}^{i=n} a_i x^{n - i}\]](http://latex.artofproblemsolving.com/0/0/5/0050577825cfc0208f72081efbfa8df9db902378.png)
be a polynomial of degree with integral coefficients. If 
and 
are odd, prove that 
has no rational roots.
and 
are given that are, respectively, the circumcenter of a triangle 
,
,
on 
.
perpendicular to 
at 
and the circle 
that inersects 
.
intersects 
at 
.