[/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 a point on side 
be a point on side 
such that lines 
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 
lies on the arc 
.
.
Y by Adventure10, Mango247
Let 
be a point in the interior of the triangle 
. The lines 
, 
, and 
intersect the sides 
, 
, and 
at 
, and 
, respectively. A point 
is taken on the ray 
such that ![$E\in [BQ]$](//latex.artofproblemsolving.com/6/e/7/6e7f283bf836f5e63e34dec2cfd3b42d3dad2ce7.png)
and 
. If 
and 
are perpendicular, and 
, prove that 
.
be a point in the interior of the triangle 
. The lines 
, 
, and 
intersect the sides 
, 
, and 
at 
, and 
, respectively. A point 
is taken on the ray 
such that ![$E\in [BQ]$](http://latex.artofproblemsolving.com/6/e/7/6e7f283bf836f5e63e34dec2cfd3b42d3dad2ce7.png)
and 
. If 
and 
are perpendicular, and 
, prove that 
.
.
.
are harmonic rays. So 
.
are harmonic rays. So 
.
.
.
.
are collinear.
.
.
and similarity ratio is 
.
and 
so 
.
