AoPS Academy
over 
. Ignore "pathological" solutions.[/quote]
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 
. At the ends of the chord to the circle are drawn tangent and equal segments 
and 
, lying on different sides wrt line . Prove that line divides the segment 
in half.
is positive, then ![\[ \log_e (1 + 1/x) > 1 / (1 + x).\]](http://latex.artofproblemsolving.com/f/3/0/f30f05414aace0d4c5ce8549afb430bfff34ccfc.png)
