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be a positive integer. A Japanese triangle consists of 
circles arranged in an equilateral triangular shape such that for each 
, 
, 
, , the 
row contains exactly 
circles, exactly one of which is coloured red. A ninja path in a Japanese triangle is a sequence of circles obtained by starting in the top row, then repeatedly going from a circle to one of the two circles immediately below it and finishing in the bottom row. Here is an example of a Japanese triangle with 
, along with a ninja path in that triangle containing two red circles., find the greatest 
such that in each Japanese triangle there is a ninja path containing at least red circles.
, such that 
and 
.
such that 
is prime, 
with 
and ?
be a function such that for every regular 
-gon 
we have 
. Prove that 
for all reals 
.
.So 
lies on 
and 
and 
.Let the angle bisector of 
cuts the line 
at 
.Now 
can be easily found and 
.So it's easy to show that 
. So 
.Now 
is the medial triangle of 
and the cevian 
is also the angle bisector of 
. Hence the result follows.
cut circumcircle 
of again at 
is feet of perpendicular from 
to 
Let 
be feet of perpendiculars from to 
is Simson line with pole . 
is kite 
. By Archimedes theorem, 
for 
and 
for 
is internal bisector of . Likewise, 
are internal bisectors of 
concurrent at incenter of 
.
are the midpoints of the sides 
.
,
.
are concurrent.
