over 
.
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 samrocksnature, Mango247
The square kingdom of Dania with a side of 
km consists of two rectangular (non-square) counties Anya and Bania, and two square counties, Vania and Gania. Neither side of these counties exceeds 
km. Prove that the rectangles Anya and Bania are equal.
Oral given note: Square counties, Vania and Gania, have different sizes.
km consists of two rectangular (non-square) counties Anya and Bania, and two square counties, Vania and Gania. Neither side of these counties exceeds 
km. Prove that the rectangles Anya and Bania are equal.Oral given note: Square counties, Vania and Gania, have different sizes.
This post has been edited 3 times. Last edited by parmenides51, Apr 14, 2021, 8:25 PM
Reason: added note, see @6
Reason: added note, see @6
,
a quadrilateral region including the boundary and inside region.
