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be an acute triangle. Points 
, and 
lie on a line in this order and satisfy 
. Let 
and 
be the midpoints of 
and 
, respectively. Suppose triangle 
is acute, and let 
be its orthocentre. Points 
and 
lie on lines 
and 
, respectively, such that 
and are concyclic and pairwise different, and 
and are concyclic and pairwise different. Prove that 
and are concyclic.
+5 w, let 
be all the positive integers smaller than that are coprime to . Find all 
such that![\[gcd(N, c_i + c_{i + 1}) \neq 1\]](http://latex.artofproblemsolving.com/1/0/7/1075290d442e8c1f54237aad8ad07126f455f1f1.png)
for all 
.
of positive integers is called central if for every positive integer 
, the arithmetic mean of the first 
terms of the sequence is equal to .
, 
, 
, 
of positive integers such that for every central sequence 
, 
, 
, , there are infinitely many positive integers with 
.
+2 w
such that 
be a set of Natural numbers from 
to 
)