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)
.
.
.
that satisfy the equation 
.
intersect 
Circles 
and 
are internally tangent to 
and 
respectively, and both are tangent to 
Similarly, circles 
and 
are externally tangent to 
and 
respectively.
the segment 
has a length of 
and 
find the length of segment 
and 
Prove that
.
Prove that
and 
Prove that
,
,
is not greater than 
.
.
.
Y by
On side 
of triangle 
, point 
is selected. A straight line passing through intersects the segment 
at point 
and the ray 
at point 
. The circumscribed circle of the triangle 
intersects 
, the circumscribed circle of the triangle , at points 
and 
. Straight lines 
and 
intersect with for the second time at points 
and 
, respectively. Prove that 
.
of triangle 
, point 
is selected. A straight line passing through intersects the segment 
at point 
and the ray 
at point 
. The circumscribed circle of the triangle 
intersects 
, the circumscribed circle of the triangle , at points 
and 
. Straight lines 
and 
intersect with for the second time at points 
and 
, respectively. Prove that 
.This post has been edited 1 time. Last edited by parmenides51, Mar 15, 2020, 4:24 PM
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