integers 
,
,
is tangent to the circle 
at 
.
and 
cut the circle at 
and 
respectively. 
is the in-centre of 
,
.
and 
intersect at 
,
and 
intersect at 
.
lie on a circle.
be the number of positive divisors of 
.
be the number of positive divisors of 
when divided by 
.
.
![$f : [0,1] \to \mathbb{R}$](http://latex.artofproblemsolving.com/4/9/b/49bb7defda1cef96adf247814e557ab43137c8a6.png)
continuous such that 
, 
and 
.![$P \in \mathbb{Q}[X]$](http://latex.artofproblemsolving.com/7/9/0/790ff8362f092209912705870106322c0214bfec.png)
such that :![\[ \forall x \in [0,1] : |f(x)-x^{m}P(x)| \leq u \]](http://latex.artofproblemsolving.com/1/6/6/16689ab8bef19c9ab8fa71b1c6e8d27d85fe6a3f.png)
![$(P_{n})_{n\geq 1} \in \mathbb{Q}[X]^{\mathbb{N}}$](http://latex.artofproblemsolving.com/a/c/e/ace9070d3d02a04bf28729b20a10c6496b0ff966.png)
such that 
for all 
and an extractrice 
such that :![\[ \forall x \in [0,1] , n \geq 1, |P_{n}(x)-S_{\phi(n)}(x)| \leq \frac{1}{n} \]](http://latex.artofproblemsolving.com/9/d/0/9d05f2b5d7de79074d74c30dddfd9a3bcb06539e.png)
converge uniformely to 
in ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
![$f \in C([0,1];[0,1])$](http://latex.artofproblemsolving.com/4/7/5/475518b7db0b787dd90b93f602c7b912c663fba3.png)
bijective, ![$(a_k) \in [0,1]^\mathbb N$](http://latex.artofproblemsolving.com/a/d/8/ad8a80d847c60ee8e916a1f9b08b7b03ec17c814.png)
with 
converge.
converge?
and 
.
with multiplication of matrixes operation is making an isomorphic-group structure with 
.
.
converge?
be a monic, real polynomial of degree 
;
for some real coefficients 
.
for all integers 
such that 
.
for which 
.
derivatives at each point of 
.
,
,
,
there is a number 
in the open interval 
for which
be an integer, show that there are infinite perfect squares of the form 
Y by Adventure10, Mango247, Mango247, Mango247
Let 
be one of the two points of intersection of two circles with centers 
respectively.The tangents at to the two circles meet the circles again at 
. Let a point 
be located so that 
is a parallelogram. Show that is also the circumcenter of triangle 
.
be one of the two points of intersection of two circles with centers 
respectively.The tangents at to the two circles meet the circles again at 
. Let a point 
be located so that 
is a parallelogram. Show that is also the circumcenter of triangle 
.
.
the second intersection point of 
.
is perpendicular to 
and 
respectively.(because AB,AC,AQ are the radical axes of 
,
.
is perpendicular at 
,
.
(because of the parallelogram).So it suffices to prove that 
which is true because these angles have their lines perpendicular.
.
then 
lies on perpendicular bisector of ![$ [AC]$](http://latex.artofproblemsolving.com/5/b/0/5b08b2b92472414250205866b93405af4772e86c.png)
,
.
,
,
is the perpendicular bisector of 
is also perpendicular bisector of 
,
, let 
and 
we get 
is parallelogram we have 
, hence we get 
