[/quote]
,
which have the same digits, but in a different order. (We forbid leading zeros.)
be a prime number and let 
be an integer with 
Prove that there exist integers 
and 
with 
and![\[ \left \{\frac{sm}{p} \right\} < \left \{\frac{sn}{p} \right \} < \frac{s}{p} \]](http://latex.artofproblemsolving.com/9/a/b/9ab00a059f9f1119a6b8a698e5ed713f19163f89.png)
.
a real number, let 
denote the greatest integer less than or equal to 
denote the fractional part of x.
,
.
with integer coefficients such that for any positive integer 
,
is positive and 
is recursively defined such that 
is not a power of 
is even, then 
is the largest odd factor of 
where 
for all 
.
of integer numbers, we define 
as the sum of the elements of 
and 
whose intersection is empty, whose union is the set 
and such that the product 
is a perfect square.
,
or 
,
divides 
?
in triangle 
.
,
,
be the feet of the internal angle bisectors of 
,
,
respectively.
be the incenter of triangle 
,
be the foot of the perpendicular from 
.
is cyclic, then the center of its circumscribed circle lies on segment 
.
board. Julia
where 
and 
are integers.
and![\[(x-y)[f(x)+f(y)]\leqslant f(x^2-y^2), \forall x,y \in \mathbb{R}\]](http://latex.artofproblemsolving.com/8/0/b/80bb1d7a992c9683a0c2a07b2febe6271f4e60f5.png)
and 
Prove that
,
be the orthocenter of it and 
be any point on the side 
.
are on the segments 
,
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
.
and 
where 
.
is called 
if there exists a permutation 
of the numbers 
,
and 
have different parities for every 
;
the sum 
is a quadratic residue modulo 
.
Y by Dilshodbek, Adventure10
Suppose that 
is a parallelogram such that 
. Let the point 
to be on 
such that 
is perpendicular to . Let the point 
to be the midpoint of 
. Let the point 
to be the intersecting point of the line 
with the circumcircle of 
. Prove that 
is concyclic.
is a parallelogram such that 
. Let the point 
to be on 
such that 
is perpendicular to . Let the point 
to be the midpoint of 
. Let the point 
to be the intersecting point of the line 
with the circumcircle of 
. Prove that 
is concyclic.
and 
denote the circles through 
and through 
respectively. Let 
be the intersection point of 
and 
.
we have 
is the radical axis of 
the point of intersection of 
and 
.
also that 
then 
locates a point 
such that 
is parallel to 
then 
also 
hence it is easy to see 
also that 
.
hence it is easy to see 
then 
then 
is cyclic as 
is cyclic then 
is cyclic
then it is easy to see that 
is cyclic.
over 
to 
Applying Reim Theorem gives 
is also cyclic 
is cyclic
