[/quote]
,
then prove \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}+\frac{1}{2}\cdot\frac{abc}{ab+bc+ca}\ge \frac{5}{3}.$$
and 
,
cookies from one of the piles, of which he eats 
and puts 
for which Marica has a winning strategy.
be non-negative real numbers such that 
Prove that
be the circumcenter, symmedian point of 
.
at 
at 
,
at 
are concurrent.
and 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
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 
such that lines 
and 
are parallel. Let 
be an interior point of 
.
and 
meet side 
,
and 
intersect at a point 
.
,
are collinear.
Y by
Draw a line containing the angle bisector 
of the right triangle 
, with right angle at C . The tangent drawn at point 
to the circle circumcscribed of the triangle intersects that line at point 
, and 
is the symmetric point of wrt point . Let 
be the intersection point of the lines 
and 
. Prove that the lines 
and 
are perpendicular.
of the right triangle 
, with right angle at C . The tangent drawn at point 
to the circle circumcscribed of the triangle intersects that line at point 
, and 
is the symmetric point of wrt point . Let 
be the intersection point of the lines 
and 
. Prove that the lines 
and 
are perpendicular.
