[/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 
.
)
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 Adventure10, Mango247
Let 
be an isosceles triangle with 
. On the extension of the side 
we consider the point 
such that 
. The perpendicular bisector of the segment 
meets the internal and the external bisectors of the angle 
at the points 
and 
, respectively. Prove that the points 
are concyclic.
be an isosceles triangle with 
. On the extension of the side 
we consider the point 
such that 
. The perpendicular bisector of the segment 
meets the internal and the external bisectors of the angle 
at the points 
and 
, respectively. Prove that the points 
are concyclic.
and 
.
,
and therefore 
,
is cyclic. Since 
and 
is right, we have that 
,
too, so 
is cyclic, as desired.
passes through the mid point of 
)
is isoceles. We know that the perpendicular bisector of 
passes throught the mid point of 
)
is cyclic. Since 
is the midsegment of 
,
is cyclic.
is cyclic.
