[/quote]
,
be a positive integer. There are two numbers written on a blackboard, one red and one blue. Initially, both are 0. We define the following procedure: at each step, we choose a nonnegative integer 
(not necessarily distinct from the previously chosen ones), and, if the red and blue numbers are 
and 
respectively, we replace them with 
and 
,
.
are positiv real number, such that 
prove inequality 
and 
.
be the set of all positive integers. Determine all subsets 
of 
such that
and 
intersect at 
.
be the circumcircle of the triangle 
;
and 
,
and 
intersects 
.
on 
.
and 
meet at 
,
.
.
?
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 
,
,
and 
.
is prime, 
with 
and 
Y by Adventure10, Mango247
Consider a circle 
. The point 
lies outside and a line is drawn through , cutting at distinct points 
and 
. Circles 
and 
are drawn through which are tangent to at and respectively.Prove that the difference of the radii of and is independent of the positions of ,
and .
. The point 
lies outside and a line is drawn through , cutting at distinct points 
and 
. Circles 
and 
are drawn through which are tangent to at and respectively.Prove that the difference of the radii of and is independent of the positions of ,
and .
and 
are joined.
are tangent to each other, 
are collinear.
is a parallelogram.
which is independent of the positions of 
(proved)
and 
which are both radii.
and similarly 
(radii) and so 
(
