[/quote]
,
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 
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.
Y by Adventure10, Mango247
Let 
be a triangle each of whose angles is greater than 
. Suppose a circle centered with 
cuts segments 
in 
in 
and 
in 
such that they are on a circle in counterclockwise direction in that order.Suppose further 
are equilateral. Prove that:
The radius of the circle is 
where 
is area.
be a triangle each of whose angles is greater than 
. Suppose a circle centered with 
cuts segments 
in 
in 
and 
in 
such that they are on a circle in counterclockwise direction in that order.Suppose further 
are equilateral. Prove that:
The radius of the circle is 
where 
is area.
we have 
but since 
is an isosceles trapezoid with legs 
then 
i.e. 
is antiparallel to 
Assuming that 
separate 
let 
constructed outside 
Then 
(1) 
i.e. 
are isogonals WRT 
Hence, we deduce that 
of 
i.e. the 1st Isodynamic point of 
we have 
(2). Thus, denoting 
the radius of the subject circle, from (1) and (2) we get
we get
![$\cos C=\frac{a^2+b^2-c^2}{2ab} \ , \ 2[\triangle ABC]=bc \sin A$](http://latex.artofproblemsolving.com/9/4/a/94a7f884caf279e03cd5a259a2833a7a253ecf3f.png)
yields![$AX^2=\frac{_1}{^2}(a^2+b^2+c^2)+2\sqrt{3}[\triangle ABC] \ \Longrightarrow \varrho= \frac{2abc}{a^2+b^2+c^2+4\sqrt{3}[\triangle ABC]}$](http://latex.artofproblemsolving.com/7/6/5/7655674673dfa159460e1e6f43daf8bae67d50fd.png)
do not separate ![$AX^2=\frac{_1}{^2}(a^2+b^2+c^2)-2\sqrt{3}[\triangle ABC] \ \Longrightarrow \varrho= \frac{2abc}{a^2+b^2+c^2-4\sqrt{3}[\triangle ABC]}$](http://latex.artofproblemsolving.com/4/1/a/41ae8f96dc6b853954080ea0592755cddc9bf1b3.png)
,
,
