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be a positive integer. Prove that there exist three permutations 
, 
, and 
of 
such that ![\[\left|\sqrt{a_k}+\sqrt{b_k}+\sqrt{c_k}-2\sqrt{N}\right|<2023\]](http://latex.artofproblemsolving.com/9/8/0/98097fb721722b9103ef65d72322956877e262cb.png)
for every 
.
such that
are all powers of 
.
be a triangle. Circle 
passes through 
, meets segments 
and 
again at points 
and 
respectively, and intersects segment 
at 
and 
such that lies between 
and . The tangent to circle 
at and the tangent to circle 
at meet at point 
. Suppose that points and are distinct. Prove that line 
is parallel to .
be the midpoint of 
Then according to the topic Easy geometry, we get that 
and 
are isosceles right triangles with commom apex 
Since 
then it follows that 
and 
are congruent by SAS 
and 
imply that 
be an arbitrary quadrilateral. Squares with centers 
are constructed on 
respectively, all outwards or all inwards. Prove that 
and 
.