,
.
lies on segment 
such that 
.
and 
be the circumcenter and incenter of triangle 
,
and 
be the circumcenter and incenter of triangle 
,
and 
intersect at point 
.
.
the number of positive integer 
satisfying 
is less than or equal to 
where 
is a constant independent of 
,
satisfies 
and 
.
to line 
meets the angle bisector of 
.
holds.
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
Prove that
and 
.
Let 
.
and 
Prove that
and 
be positive real numbers with 
.
For which 
such that 
for all 
,
.
and 
of an equilateral triangle 
are taken points 
and 
,
, points 
and 
so that 
.
and 
is equal to 
.
over 
.
Y by
1. The infinite Morse sequence of zeros and ones, 011010011001..., is constructed as follows: start with 0, then at each step, append a block of the same length as the current sequence, obtained by replacing 0 with 1 and vice versa in the existing block. Is this sequence periodic?
2. On an infinite (two-way) tape, a text in Russian is written. It is known that in this text, the number of distinct 15-symbol blocks is equal to the number of distinct 16-symbol blocks. Prove that the text on the tape is periodic in both directions (i.e., bi-infinite and periodic), for example: "...мамамыларамумамамы...".
2. On an infinite (two-way) tape, a text in Russian is written. It is known that in this text, the number of distinct 15-symbol blocks is equal to the number of distinct 16-symbol blocks. Prove that the text on the tape is periodic in both directions (i.e., bi-infinite and periodic), for example: "...мамамыларамумамамы...".
