[/quote]
,
then prove
.
?
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)
Y by Adventure10, Mango247
Let 
be an isosceles triangle, with 
. Let 
be points on 
and 
, respectively, such that 
is parallel to and 
is parallel to 
. Further, let 
be the circumcentre of . Prove that the quadrilateral 
is cyclic.
be an isosceles triangle, with 
. Let 
be points on 
and 
, respectively, such that 
is parallel to and 
is parallel to 
. Further, let 
be the circumcentre of . Prove that the quadrilateral 
is cyclic.This post has been edited 1 time. Last edited by parmenides51, Jul 21, 2019, 4:31 AM
Reason: typo in the end, the correct is CROQ, I had before CPOQ, thanks below
Reason: typo in the end, the correct is CROQ, I had before CPOQ, thanks below
and 
are congruents, For this, 
.
