[/quote]
,
be the set of real numbers. Let 
be a function such that ![\[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\]](http://latex.artofproblemsolving.com/5/d/3/5d3476fd9195d085d0030d6036a9fde1cf6756a7.png)
for every 
.
.
for every 
or 
for every 
and 
.![\[ \frac{a^2+1}{a+1}+\frac{b^2+1}{b+1}+\frac{c^2+1}{c+1} \geq 3\]](http://latex.artofproblemsolving.com/7/d/2/7d2e34116452912cd3da2925312ad500bb976f15.png)
with real coefficients of degree 
such that the following two conditions hold:
(they are not necessarily distinct);
are 
.
is monic since the coefficient of 
is one.
be a point on side 
.
and 
are points inside 
and 
are similar and quadrilaterals 
and 
are cyclic. Let lines 
and 
meet at 
and lines 
and 
meet at 
.
and 
are parallel.
which is a perfect square is of the form 
for 
be an integer. Prove that there exists a positive integer 
with the following property: For each positive integer 
,
(possibly depending on 
)
is divisible by 
.
and 
be positive divisors of 
.
is not divisible by 
.
,
.
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 
.
of positive integers with ![\[ a^p=b!+p. \]](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 
.
,
.
.
lie on segments 
respectively. Points 
are such that 
and 
are equlateral, 
and 
lie on opposite sides of 
,
and 
lie on the opposite siedes of 
.
Y by NO_SQUARES, ehuseyinyigit
Let 
be a triangle with incircle 
, which touches sides 
and 
at points 
and 
, respectively. Choose points 
and 
on the line 
such that 
and 
. Let 
be the intersection of lines 
and 
. Define 
and 
as the intersections of 
and 
with lines 
and 
, respectively. Assume that 
is the intersection of 
and 
. Prove that 
.
be a triangle with incircle 
, which touches sides 
and 
at points 
and 
, respectively. Choose points 
and 
on the line 
such that 
and 
. Let 
be the intersection of lines 
and 
. Define 
and 
as the intersections of 
and 
with lines 
and 
, respectively. Assume that 
is the intersection of 
and 
. Prove that 
.Attachments:
at 
.
ia a parallelogram. 
intersect 
at 
intersect 
at 
bisect 
.
bisects segment 
,
