[/quote]
,
be a positive integer. A Japanese triangle consists of 
circles arranged in an equilateral triangular shape such that for each 
,
,
,
row contains exactly 
circles, exactly one of which is coloured red. A ninja path in a Japanese triangle is a sequence of 
,
such that in each Japanese triangle there is a ninja path containing at least 
,
and 
.
is prime, 
with 
and 
be a function such that for every regular 
-
we have 
.
for all reals 
.
)
are the altitudes of the triangle. The bisector of 
intersects 
at 
.
be a point such that 
and 
.
passes through the midpoint of 
& 
.
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 
for which Marica has a winning strategy.
be non-negative real numbers such that 
Prove that
Y by kazakhboy, HWenslawski, Adventure10, and 1 other user
The triangle 
is inscribed in a circle 
. Inscribed in a triangle circle touchs the sides 
in a point 
. 
— the circle inscribed in a segment 
circle of , and passing through a point . Let points 
and 
— the centers of circles and an extra inscribed circle (touching side ) respectively. Prove, that lines 
and 
are parallel.
is inscribed in a circle 
. Inscribed in a triangle circle touchs the sides 
in a point 
. 
— the circle inscribed in a segment 
circle of , and passing through a point . Let points 
and 
— the centers of circles and an extra inscribed circle (touching side ) respectively. Prove, that lines 
and 
are parallel.
,
.
.
Let 
and 
,
,
.
and 
we know that 
and 
are similar. 
is cyclic. 
.
is diametr of circumcircle of 
are on a line.Let 
be midpoint of ar
and 
are on a line. 
is intersection of 
and 
.
:
SO we are done!
-circumcircle of 
-tangency point of 
and 
are collinear. Let 
.
at the point 
then 
are collinear and 
.
.
.
.
and 
is parallelogram. ![\[ \angle HAK=\angle INM=\angle HXK \]](http://latex.artofproblemsolving.com/b/e/3/be3201468bccee88920e6b3d67507a52573b2734.png)
and 
are concyclic, 
.
,
the point where 
touches 
are collinear where 
is midpoint of an arc 
.
are collinear. Let tangent to 
and let 
.
we have that quadrilateral 
is cyclic. Now we have that 
so we have 
which implies that 
is tangent to 
it's enough to prove 
.
where 
is radius of 
or using that 
we have to prove that 
.
because 
intersects again 
we know that 
so we have to prove that 
.
so what we have to prove reduces to 
.
we have that 
.
and 
so we have that triangles 
and 
are similar which gives us 
as we wanted.
with the parallel to 
is exactly our circle and is tangent to the circumcircle at the sharky point. Let 
be the foot of the altitude from 
bisects segment 
be the midarcs 
with circumcircle of 
and 
are similar since 
and 
so 
collinear so by homothety since 
we must have the desired tangency.
and let it's antipode in it be 
.
are collinear by homothety. Notice the similar triangles 
and 
so we must have the collinearity of the midpoints of 
and 
