[/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}.$$
with 
and at most one of them being zero, show that![\[\frac{1}{\max(a^2,b)}+\frac{1}{\max(b^2,c)}+\frac{1}{\max(c^2,a)}\geq\frac{27}{4}\]](http://latex.artofproblemsolving.com/9/e/1/9e14a9ef2d1a0a98006bcf716fa33c3d9ea29bd5.png)
be an integer. Find all sequences 
satisfying the following conditions:![\[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 + n;
\]](http://latex.artofproblemsolving.com/3/c/5/3c509ec2e9013e8d3be492c8eb44a7c33841b74e.png)
![\[ \text{ (b) } a_{i + 1} + a_{i + 2} + \ldots + a_{i + n} < a_{i + n + 1} + a_{i + n + 2} + \ldots + a_{i + 2n} \text{ for all } 0 \leq i \leq n^2 - n.
\]](http://latex.artofproblemsolving.com/9/7/d/97d2a467d1c0dc8594ec024c3bb9b8c87ee85b19.png)
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 
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 
'
Y by
We consider an acute triangle 
. The altitude from 
is 
, the altitude from 
in triangle 
is 
and the altitude from in triangle 
is 
.
a) Prove that the triangles and 
are similar.
b) Prove that the segment
and the corresponding segments constructed from the vertices 
and 
all have the same length.
. The altitude from 
is 
, the altitude from 
in triangle 
is 
and the altitude from in triangle 
is 
.a) Prove that the triangles
and 
are similar.b) Prove that the segment
and the corresponding segments constructed from the vertices 
and 
all have the same length.
is a cyclic quadrilateral so 
so 
similarly 
.
.
.
gives 
,
.
be the foot of the altitude from 
.
and 
,
and 
which upon substituting gives the desired equality. 
