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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 
.
lie on the circle 
, and 
is a point such that 
are tangents to . 
is a point on the extension of 
and 
is the circumcircle of triangle 
. The circle cuts again at 
and 
cuts at the point 
. Prove that 
.
lie on the circle , and is a point such that are tangents to . is a point on the extension of and is the circumcircle of triangle . The circle cuts again at and cuts at the point . Prove that .
and are cocyclic Hence 
where 
is a point on the extention of 
and because is tangent to we have 
and Thus 
. Let 
we have 
and Thus it follows that 
Hence the result ... 
