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be the set of positive real numbers.
such that there exists a function 
with the property:![\[
f(y^2f(x) + y + c) = xf(x+y^2)
\]](http://latex.artofproblemsolving.com/e/3/0/e30ee6d9f69cc43422a5c2b9813192a632d359cb.png)
for all 
.
be non-negative real numbers such that 
Prove that
are real number satisfying
show that the equation 
has at least one real root.
be real numbers. Find the ranges of such that the improper integral 
converges.
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.
does the sequence 
defined by the initial condition 
and the recursion 
have 
for all 
where 
.
where .
and 
then prove 
and 
.
is positive, then ![\[ \log_e (1 + 1/x) > 1 / (1 + x).\]](http://latex.artofproblemsolving.com/f/3/0/f30f05414aace0d4c5ce8549afb430bfff34ccfc.png)
and area 
it is true, that
we have 
So we must have 
