[/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 
;
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
A triangle 
with 
is inscribed into a circle 
. Points 
and 
are on the sides 
and 
, respectively, such that 
. Let lines 
and 
intersect at 
, and let 
be the midpoint of arc containing 
. The circle with diameter
intersects line at 
and intersects the angle bisector of 
at 
.
a) Prove that
.
b) Let
be the incenter of 
and let 
be the -excenter of 
. Prove that 
.
with 
is inscribed into a circle 
. Points 
and 
are on the sides 
and 
, respectively, such that 
. Let lines 
and 
intersect at 
, and let 
be the midpoint of arc containing 
. The circle with diameter
intersects line at 
and intersects the angle bisector of 
at 
.a) Prove that
.b) Let
be the incenter of 
and let 
be the -excenter of 
. Prove that 
.This post has been edited 1 time. Last edited by parmenides51, Oct 25, 2020, 1:09 PM
is cyclic
is the cirumcenter of 
with 
are cyclic and 
are cyclic due to miquel point.
.
are midpoints of arc 
respectively.
,
.
,
.
((SMK), (SNC) have the same radius), and we are done.
