[/quote]
,
be the set of all positive integers. Determine all subsets 
of 
such that
and 
intersect at 
and 
.
be the circumcircle of the triangle 
;
and 
,
and 
intersects 
.
on 
.
and 
meet at 
,
.
.
?
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 
.
be a triangle 
)
are the altitudes of the triangle. The bisector of 
intersects 
at 
.
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)
Y by pisgood, Adventure10, Mango247
The triangle 
is given. On the extension of the side 
we construct the point 
with 
, where 
and on the extension of the side 
we construct the point 
with 
, where 
. Let 
be the point of intersection of the diagonals of the quadrilateral 
.
Analogous we construct the point
on the extension of the side 
, where 
and 
and on the extension of the side 
we construct the point 
with 
, where 
. Let 
be the point of intersection of the diagonals of the quadrilateral 
.
Likewise we construct the point
on the extension of the side 
, where 
and 
and on the extension of the side 
we construct the point 
with 
and 
. Let 
be the point of intersection of the diagonals of the quadrilateral 
.
Show that the area of the hexagon
is equal to the sum of the areas of the triangles and 
.
is given. On the extension of the side 
we construct the point 
with 
, where 
and on the extension of the side 
we construct the point 
with 
, where 
. Let 
be the point of intersection of the diagonals of the quadrilateral 
.Analogous we construct the point
on the extension of the side 
, where 
and 
and on the extension of the side 
we construct the point 
with 
, where 
. Let 
be the point of intersection of the diagonals of the quadrilateral 
.Likewise we construct the point
on the extension of the side 
, where 
and 
and on the extension of the side 
we construct the point 
with 
and 
. Let 
be the point of intersection of the diagonals of the quadrilateral 
.Show that the area of the hexagon
is equal to the sum of the areas of the triangles and 
.
a
and 
are equal,thus 
are equal,then we must prove that the sum of area of triangles 
,
are the area of 
