[/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 StefanS, Adventure10, Mango247, and 1 other user
Let 
be an interior point of an acute triangle 
. The circles with centers the midpoints of its sides and passing through mutually intersect the second time at the points 
, 
and 
different from . Prove that is the incenter of the triangle 
if and only if is the circumcenter of the triangle .
be an interior point of an acute triangle 
. The circles with centers the midpoints of its sides and passing through mutually intersect the second time at the points 
, 
and 
different from . Prove that is the incenter of the triangle 
if and only if is the circumcenter of the triangle .This post has been edited 1 time. Last edited by Valentin Vornicu, Sep 24, 2005, 1:57 PM
is isosceles, then 
where 
are the midpoints of the triangle 
and 
are cyclic, we get that 
,
(the center line is perpendicular on the radical axis) and also 
we get that 
,
is the interior angle bisector of the 
.
.
and 
.
so 
is cyclic. In the same way 
and 
and 
,
,
.
so we are done.
,
,
and 
so 
,
and M = 
.
,
and R = 
.
and 
,
KML with respect to O (and a scale factor of 2), this implies that O is the incenter of KML as well.
,
.
