[/quote]
,
which is a perfect square is of the form 
,
.
with integer coefficients such that for any positive integer 
,
is positive and 
is called 
if there exists a permutation 
of the numbers 
,
and 
have different parities for every 
;
the sum 
is a quadratic residue modulo 
for every 
.
of positive integers with 
prime and ![\[ a^p=b!+p. \]](http://latex.artofproblemsolving.com/9/f/6/9f674dabcb364872cef4153198e2e362ae40a2fd.png)
the set of all positive integers. Find all functions 
such that for all positive integers 
and 
is nonzero and divides 
.
,
satisfying the equation 
.
,
.
lie on segments 
respectively. Points 
are such that 
and 
are equlateral, 
and 
lie on opposite sides of 
,
and 
lie on the opposite siedes of 
,
,
,
.
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For 
represents 
.
with 
.![\[
\sqrt{a + (b - c)^2} + \sqrt{b + (c - a)^2} + \sqrt{c + (a - b)^2} \geq \sqrt{3},
\]](http://latex.artofproblemsolving.com/c/b/5/cb551e6d6ce94902b7bcb78f6da210d6c27b96d5.png)
with equality if and only if 
.
and circumcircle 
.
be the midpoints of the arcs 
of 
respectively. Let 
be the point of 
.
and the midpoint 
of 
lie on a line.
.
Where 
and 
.
lie on segment 
lie on segment 
,
and 
.
,
the orthocenter of triangle 
and 
.
.
Y by
Given a cyclic quadrilateral 
with 
and 
. Lines 
and 
intersect at 
, and lines 
and 
intersect at 
. Let 
be the midpoints of sides 
, respectively. Let 
and 
be points on segment 
and 
, respectively, so that 
is the angle bisector of 
and 
is the angle bisector of 
. Prove that 
is parallel to 
if and only if 
divides into two triangles with equal area.
with 
and 
. Lines 
and 
intersect at 
, and lines 
and 
intersect at 
. Let 
be the midpoints of sides 
, respectively. Let 
and 
be points on segment 
and 
, respectively, so that 
is the angle bisector of 
and 
is the angle bisector of 
. Prove that 
is parallel to 
if and only if 
divides into two triangles with equal area.
direction but not pretty sure with the 
direction, can anyone help me?
is a parallelogram and 
.
iff 
.
and 
.
and 
.![\[ \frac{EX}{GX} = \frac{ES}{SG} = \frac{HT}{TF} = \frac{HY}{FY}. \]](http://latex.artofproblemsolving.com/e/0/c/e0c170e9bf04bfbd87a4e72b3b7f2f1c93858b3b.png)
Again, by the similarity this means ![\[ \frac{AB}{CD} = \frac{DA}{BC} \quad \Rightarrow \quad AB\cdot BC = CD\cdot DA, \]](http://latex.artofproblemsolving.com/c/9/0/c90aaa3c7f32f7fdd231d270e38b43fc934e2ce2.png)
which essentially means the areas of 
are equal.
direction.
