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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 .![$[BH]$](http://latex.artofproblemsolving.com/a/9/f/a9fd89d48264ff1c3f9b77f62ef66f916978d99d.png)
dropped onto the hypotenuse of a triangle intersects the bisectors ![$[AD]$](http://latex.artofproblemsolving.com/0/f/3/0f3e4c424371b27673db323ced8ef0777940c0d4.png)
and ![$[CE]$](http://latex.artofproblemsolving.com/e/a/8/ea8eadb930f32cceb220dee99277c24b202fb80c.png)
at 
and 
respectively. Prove that the line passing through the midpoints of the segments ![$[QD]$](http://latex.artofproblemsolving.com/2/b/2/2b24b6b34a55e29ef31a8cc9a72f116d3896a69f.png)
and ![$[PE]$](http://latex.artofproblemsolving.com/7/9/2/792976015f6d59bd3b89b6d2d53901be85cecd92.png)
is parallel to the line .
dropped onto the hypotenuse of a triangle intersects the bisectors and at and respectively. Prove that the line passing through the midpoints of the segments and is parallel to the line .
are isosceles, hence if 
are the midpoints of , respectively , we have 
, 
, i.e. 
is cyclic, 
being the incenter. From 
. From we get 
, or 
.
and 
and 
.After that it is just simple calculations and not much thinking.
