[/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 
.
Equality holds when 
or 
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.
Y by centslordm, Mango247, Mango247
A segment 
is given on the plane and an arbitrary point 
is given on it. Using segments 
and 
as sides, are drawn the squares 
and 
, lying on the same side of . Let 
be the point of intersection of lines 
and 
. Prove that for any position of the point on segment , each straight line 
passes through some point 
, common for all such lines.
is given on the plane and an arbitrary point 
is given on it. Using segments 
and 
as sides, are drawn the squares 
and 
, lying on the same side of . Let 
be the point of intersection of lines 
and 
. Prove that for any position of the point on segment , each straight line 
passes through some point 
, common for all such lines.This post has been edited 1 time. Last edited by parmenides51, May 17, 2021, 11:15 AM
.
cuts the line 
.
.
and 
give us the fixed point 
.
