[/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
In the figure, 
and 
are isosceles triangles (
) and the angles 
and 
have measures 
.
a) Using geometric properties, calculate the measure of angle
.
b) Knowing that
, calculate the length of segment 
.
c) Calculate the length of segment
.
and 
are isosceles triangles (
) and the angles 
and 
have measures 
.a) Using geometric properties, calculate the measure of angle
.b) Knowing that
, calculate the length of segment 
.c) Calculate the length of segment
.
= 
Therefore 
![[asy]
import olympiad;
size(4cm);
pair A=(0,0),B=dir(0),C=dir(36),D=dir(72),E=(2cos(0.4pi),0);
dot(A^^B^^C^^D^^E);
draw(A--B--C--cycle);
draw(A--D--E);
draw(D--C,dotted);
label("A",A,dir(-90));
label("B",B,dir(-90));
label("C",C,dir(40));
label("D",D,dir(90));
label("E",E,dir(-90));
[/asy]](http://latex.artofproblemsolving.com/8/1/d/81d844532c6322f876557ea3cc299e01fa720376.png)
is simply triangle 
since it is isosceles. Also, 
,
.
gives 
.
