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.
has 
obtuse. Let 
, 
, 
be the feet of the altitudes from 
, 
, 
respectively. Let 
, 
, 
be arbitrary points on 
, 
, 
respectively. The circles with diameter 
, 
, 
are drawn. Show that the lengths of the tangents from the orthocentre of 
to these circles are equal.
Prove that
Where 
Prove that
Let 
Prove that
Let 
Prove that
is inscribed in a circle. Given 
and 
. Let 
and 
be the orthocentre and incentre of respectively. Find the ratio of 
to 
.
is tangent to the sides , , 
of a triangle at points 
respectively. The lines 
and 
meet 
at points 
and 
. Prove that the circumcircle of triangle 
and points and lie on a line.
such that the polynomial 
has 8 linear factors of the form 
with 
for 
. Find all possible values of the constant 
.
be real numbers such that ![\[ab^2+bc^2+ca^2=6\sqrt{3}+ac^2+cb^2+ba^2.\]](http://latex.artofproblemsolving.com/3/d/0/3d015292c249a69a44d5f923092201d7a271b896.png)
Find the smallest possible value of 
.
positive integers, less than or equal to 
, such that the difference between any 
elements in the set is at least 
.
, that is, with 
nonzero, let 
be the product 
.
and 
. with at least one 
amoung their digits satisfy that is a power of 2?
such that 
.
must be either powers of 
or 
work, where each element is either a fixed 
or 
,
