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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.
+4 w
there are elements 
in an arithmetic progression in the order 
such that 
.
we have![\[\max(|\sin x|,|\sin (x+1)|)>\frac13.\]](http://latex.artofproblemsolving.com/e/d/e/ede9b524370c457fe41038cca5663e3c9bdaf320.png)
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 
.
be odd positive integers such that 
and both 
and 
are powers of 2. Prove that 
be an acute triangle with circumcenter 
and orthocenter , satisfying 
. The tangent line at 
to the circumcicle of intersects 
in 
. Let 
be the midpoint of 
. Prove that 
.
be midpoint of side . It is well-known that 
is parallelogram. is orthocenter of triangle 
. Since 
and 
, we deduce that 
. 
and 
.
![$ a,b,c\in [0,1] $](http://latex.artofproblemsolving.com/2/1/b/21b329d0a92c18c6a2861742177969d5977d4558.png)
. Prove that
and 
Prove that
be real numbers so that 
and 
Show that
