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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 
.
and 
and a fixed point 
equidistant from these lines. What is the locus of projections 
of on 
, where 
is on 
, 
on , and angle 
is right?
and the line 
.
line 
.
.
(of the projection of on ), checked it via geogebra and is correct
and 
being parallel. (see https://www.ommenlinea.org/wp-content/uploads/concursos/nacionales/nac01.pdf).
and and a fixed point equidistant from these lines. What is the locus of projections of on , where is on , on , and angle is right?
.
and 
be the feet of perpendiculars from 
:![\[\measuredangle XYM = \measuredangle PBM = 90 - \measuredangle MAP = 90 - \measuredangle MXY,\]](http://latex.artofproblemsolving.com/4/3/9/439d7bf4f82d4bf3cafb4026be0246134028eb86.png)
so 
,