AoPS Academy
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 
.
be a center of circle which passes through vertices of quadrilateral 
, which has perpendicular diagonals. Prove that sum of distances of point to sides of quadrilateral is equal to half of perimeter of .
be a center of circle which passes through vertices of quadrilateral , which has perpendicular diagonals. Prove that sum of distances of point to sides of quadrilateral is equal to half of perimeter of .
be the intersection of the diagonals and 
be the midpoints of 
and 
respectively. Just use some angle chasing to prove 
is a parallelogram.