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be nonnegative real numbers such that 
Find the minimum value of 
be positive reals such that 
Find the maximum value of ![$$x_1+\sqrt{x_1x_2}+\sqrt[3]{x_1x_2x_3}+\sqrt[4]{x_1x_2x_3x_4}+\sqrt[5]{x_1x_2x_3x_4x_5}.$$](http://latex.artofproblemsolving.com/2/5/f/25f59ee5503bb04b51590698e5fe2b8352828ddc.png)
. Let 
be a subset of 
such that no sum of three distinct elements of is divisible by 
. What is the maximum value of 
.
that satisfy the equation 
.
be a circle, and let a line 
intersect at two points, 
and 
Circles 
and 
are internally tangent to at points 
and 
respectively, and both are tangent to at a common point 
Similarly, circles 
and 
are externally tangent to at and respectively, and are tangent to at points 
and 
respectively. is 
the segment 
has a length of 
and 
find the length of segment 
,
,
, 
, 
is not greater than 
.
, the perpendicular from onto 
meets at 
, the perpendicular from 
onto 
meets at . The length of 
is , the length of 
is 
, the length of 
is 
and the length of 
is 
. Find .
and 
Prove that
.
Prove that
and 
Prove that
.
.
