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where those are complex numbers satisfying
and 
?
, let 
be the number of pairs of integers 
such that the number 
is a perfect square. Prove that the limit
exists and find its value.
with 
and let 
.![\[P\left(\frac{p_1}{1-p_1}+\frac{p_2}{1-p_2}+\ldots+\frac{p_n}{1-p_n}\right).\]](http://latex.artofproblemsolving.com/9/8/f/98f9aaa76faf8ff093842f1cfe25040e9fa23da1.png)
has 
, 
, 
, and 
. If 
is the intersection of 
and 
, what is the measure, in degrees, of 
?
be the unique positive real number satisfying ![\[ 2^x+32^x=8^x+16^x. \]](http://latex.artofproblemsolving.com/2/5/8/258d9eaddb9352a4f72951eff657d8559553d722.png)
Compute 
.
denote the product of all positive integers 
such that the least common multiple of 
and 
is 
. Compute the remainder when is divided by 
.
be a triangle with 
and 
. Let 
be the foot of the altitude from 
to 
, and let 
be the midpoint of 
. If 
, compute 
.
be an acute triangle (
) inscribed in a circle with center 
. Let 
, 
, and 
be the altitudes of triangle (
, 
, 
). The altitudes 
and 
of triangle 
intersect at point 
(
, 
).
.
and 
be the feet of the perpendiculars from point 
to lines and , respectively. Prove that 
.
and 
be points on line 
such that the lines 
and 
are perpendicular to line 
. Prove that the three lines 
, 
, and are concurrent.
but I have no idea how to prove it
, define 
to be the minimum value of the sum ![\[ \sum_{k=1}^n \sqrt{(2k-1)^2+a_k^2}, \]](http://latex.artofproblemsolving.com/e/6/7/e677cefac5330430da1267d7004dee9c81a5e071.png)
where 
are positive real numbers whose sum is 17. There is a unique positive integer for which is also an integer. Find this .
![\[x=\frac{4}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)(\sqrt[16]{5}+1)}.\]](http://latex.artofproblemsolving.com/8/b/a/8baf05a34977902b41253e7ce7e3f3c89282f6eb.png)
Find 
.
,
,
,
.
and its absolute and relative error, known that its absolute error is equal or lower than 
