[/quote]
,
be non-negative real numbers such that 
Prove that
be the circumcenter, symmedian point of 
.
at 
,
at 
,
at 
are concurrent.
be distinct natural numbers each of which has exactly 
positive integer factors. For instance, 
has exactly 
.
divides any of the 
'
be an integer. Prove that 
can be written as a polynomial with integer coefficients in the variables 
,
and 
.
,
are real numbers such that 
is an integral multiple of 
.
,
for all positive integers 
.
be the orthocenter of an acute-angled triangle 
.
centered at the midpoint of 
and passing through 
and 
.
,
,
and 
.
with real coefficient. It is known that 
and 
.
can't have 
distinct real roots.
be an acute triangle. Let 
be a point on side 
be a point on side 
such that lines 
and 
are parallel. Let 
be an interior point of 
.
and 
meet side 
and 
intersect at a point 
.
,
are collinear.
.
,
as the greatest exponent 
such that 
divides 
.
and infinitely many 
such that 
.
and 
.
is given. Let 
be the intersection point of 
lies on the arc 
.
.
Y by
Let 
be a parallelogram whose diagonals meet at 
. Denote by 
the midpoint of 
. Let 
be a point such that 
is tangent to the circumcircle of 
and 
is tangent to the circumcircle of 
. Prove that points 
are collinear.
(Patrik Bak, Slovakia)
be a parallelogram whose diagonals meet at 
. Denote by 
the midpoint of 
. Let 
be a point such that 
is tangent to the circumcircle of 
and 
is tangent to the circumcircle of 
. Prove that points 
are collinear.(Patrik Bak, Slovakia)
;
and 
as 
and 
are isosceles trapezoids. Also, let 
be the midpoint of 
;
.
is cyclic.![\[\measuredangle BQA = \measuredangle QBA + \measuredangle QAB = \measuredangle BXM + \measuredangle MXA = \measuredangle BXA. \ \ \blacksquare\]](http://latex.artofproblemsolving.com/2/a/b/2abed00439c1b9a1cce37e38ff41cd0a30cf8458.png)
But 
is an isosceles trapezoid and thus cyclic, hence 
are all concyclic. It follows that 
which implies the result.
