and 
then prove 
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
and 
.
are real number satisfying
show that the equation 
has at least one real root.
be real numbers. Find the ranges of 
converges.
converge?
,where 
,
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 
and 
are odd, prove that 
has no rational roots.
is positive, then ![\[ \log_e (1 + 1/x) > 1 / (1 + x).\]](http://latex.artofproblemsolving.com/f/3/0/f30f05414aace0d4c5ce8549afb430bfff34ccfc.png)
does the sequence 
defined by the initial condition 
and the recursion 
have 
for all 
Y by Pluto1708, Adventure10, Mango247
In the triangle 
its incircle with center 
touches its sides 
and 
in the points 
respectively. Through is drawn a line 
. The points 
and 
are reflections of with respect to the line . Prove that the lines 
and 
intersects at a common point.
its incircle with center 
touches its sides 
and 
in the points 
respectively. Through is drawn a line 
. The points 
and 
are reflections of with respect to the line . Prove that the lines 
and 
intersects at a common point.This post has been edited 1 time. Last edited by parmenides51, May 29, 2019, 7:11 PM
Reason: typo, edited in last line
Reason: typo, edited in last line
,
,
are parallel to each other because they are all perpendicular to 
is the correct, instead of 
,
be the complex numbers corresponding to 
We also have that 
We have that 
lies on all three lines 
.
and 
are concurrent at a point at infinity
