[/quote]
,
.
be the feet of the altitudes from 
to 
to 
respectively.
and 
as the reflections of 
and 
be the circumcircle of 
.
with 
the intersection of ray 
with 
is the circumcenter of 
can be inscribed within a circle.
be a triangle whose incircle is tangent to 
,
,
at 
,
and 
be the midpoints of 
and 
,
and 
lie on 
so that 
and 
.
and 
intersect on the circumcircle of 
,
lie on a line in this order and satisfy 
.
and 
,
is acute, and let 
and 
,
and 
and 
and 
.
intersect the circumcircle of 
be the midpoint of 
and let the perpendicular bisector of 
intersect 
at 
.
.
be an increasing sequence of nonnegative integers such that every nonnegative integer can be expressed uniquely in the form 
,
and 
are not necessarily distinct. Determine 
.
be a cyclic quadrilateral with 
.
not containing 
and 
.
,
.
and find out the set of numbers 
(unordered, of course). Find the minimal number of questions, which are necessary to find out all numbers 
.
and 
.
?
be the number of positive integers that divide the integer 
.
,
?
is a square of an integer.
be all positive integers smaller than 
such that 
for all 
is the largest positive integer that divides both 
and 
.
.
,
,
are odd primes and![\[2^ap^b=(p+2)^c-1.\]](http://latex.artofproblemsolving.com/5/8/5/58518a8ebed14836b72af743ff100eae2efba5e2.png)
let 
be the circumcircle of 
and let 
be its radius. Point 
such taht 
and point 
is the foot of the perpendicular from 
mets 
such that 
and 
lie on a line in that order. Finally, let 
and 
.
Y by Rounak_iitr
In 
, 
, point 
lies on side 
, 
and 
are the centers of the circumcircles of 
and 
, respectively. Lines 
and 
intersect at point 
. If 
is the incenter of and 
is the orthocenter of 
, then prove that the four points 
are on the same circle.
, 
, point 
lies on side 
, 
and 
are the centers of the circumcircles of 
and 
, respectively. Lines 
and 
intersect at point 
. If 
is the incenter of and 
is the orthocenter of 
, then prove that the four points 
are on the same circle.
this implies 
cyclic 
and 
so 
is Concyclic
be the orthocenter of 
,
.
is cyclic. In addition, we have 
so 
is cyclic, i.e. 
is a cyclic pentagon. 
would still be cyclic, however, as this configuration yields 
