[/quote]
,
undergoes the following procedure in order to obtain the number 
:
;
;
.
.
,
,
,
.
.
show that there exists a prime 
such that one can choose distinct integers 
such that p divides 
for all 
.
and a positive integer 
is given.
and for each 
![\[f_n(k)=\frac{a_n+\dots+a_{k+1}}{n-k}-\frac{a_k+\dots+a_1}{k}\]](http://latex.artofproblemsolving.com/0/4/1/041f80cf63f17c0f62c166021af474a5d305eb87.png)
Then for every integer 
the condition![\[a_{n+1}=\max_{0\leq k<n}(f_n(k))\]](http://latex.artofproblemsolving.com/b/1/f/b1f5aff787de8fc9c113cf8441a6b2900a19f24f.png)
is satisfied. Prove that this sequence must be eventually constant.
is a pentagon whose vertices lie on circle 
where 
.
and 
intersect at 
,
meet 
at 
.
is the midpoint of arc 
on 
.
holds, then what is the value of 
?
and 
and 
is a prime number
of positive integers such that 
is an integer.
be an acute triangle and 
its orthocenter. Let 
be a point on the parallel through 
to 
such that 
.
and 
as points on the parallels through 
to 
are positioned around the sides of 
,
it's circumcenter. Let 
and 
the intersection of the circumcircle of 
,
is the reflection of 
,
and the parallel through 
,
and the perpendicular through 
and 
a point on 
,
is the midpoint of 
.
lie on a line.
be an integer. Find all sequences 
satisfying the following conditions:![\[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 + n;
\]](http://latex.artofproblemsolving.com/3/c/5/3c509ec2e9013e8d3be492c8eb44a7c33841b74e.png)
![\[ \text{ (b) } a_{i + 1} + a_{i + 2} + \ldots + a_{i + n} < a_{i + n + 1} + a_{i + n + 2} + \ldots + a_{i + 2n} \text{ for all } 0 \leq i \leq n^2 - n.
\]](http://latex.artofproblemsolving.com/9/7/d/97d2a467d1c0dc8594ec024c3bb9b8c87ee85b19.png)
be all positive integers smaller than 
such that 
for all 
is the largest positive integer that divides both 
.
,
intersects 
and 
for the second time, respectively. Similarly, circle 
intersects 
at 
and 
,
intersects 
and 
at 
and 
for the second time, respectively. Let 
and 
be circumcenters of triangle 
and 
,
are collinear.
and any 
,
are also in 
for positive integers 
be an isosceles triangle with 
.
be a point on 
.
be a point inside the triangle such that 
and![\[
CL \cdot BD = BL \cdot CD.
\]](http://latex.artofproblemsolving.com/1/6/7/16771838c95f86e92f79b8d049e46ab473e6287d.png)
Prove that the circumcenter of triangle 
lies on line 
.
satisfying 
and
for all 
.
Y by Adventure10, Rounak_iitr, and 1 other user
Let 
be a cyclic convex quadrilateral. Let 
be the intersection of lines 
. 
is the intersection of line passing 
and perpendicular to 
, and line passing 
and perpendicular to 
. 
is the intersection of line passing 
and perpendicular to , and line passing 
and perpendicular to 
. Prove that three points 
are collinear.
be a cyclic convex quadrilateral. Let 
be the intersection of lines 
. 
is the intersection of line passing 
and perpendicular to 
, and line passing 
and perpendicular to 
. 
is the intersection of line passing 
and perpendicular to , and line passing 
and perpendicular to 
. Prove that three points 
are collinear.
,
,
.
,
,
.
are collinear.
,
.
and 
are similar, hence 
.
to 
.
,
,
and 
.
,
is cyclic. But 
is the power of point 
,
.
)
,
are collinear, as desired.
with center 
which takes 
.
to 
so it sends 
to 
with center 
to 
.
to 
so it sends 
to 
or![\[{\overline{EM}\over \overline{EA}} ={\overline{EB}\over \overline{EL}}\]](http://latex.artofproblemsolving.com/6/f/6/6f6f2659af985943f0d47f840013121e050e8730.png)
![\[ \overline{EM} \cdot \overline{EL} = \overline{EA}\cdot \overline{EB} (=\overline{EC}\cdot \overline{ED}) \]](http://latex.artofproblemsolving.com/2/f/7/2f7921adb1c46973df00f5022f3fb159ca16ae3f.png)
![\[ \angle BDC = \angle BAC = \alpha \]](http://latex.artofproblemsolving.com/9/4/6/946706eb26dfa666e5d54ecdf492aadbe0b89c4c.png)
and![\[ \angle LDB = \angle ACM = \beta \]](http://latex.artofproblemsolving.com/a/1/8/a18a11c024a8562bc2060f86ea9973614e07e694.png)
![\[\angle AMC = \pi -\alpha-\beta\]](http://latex.artofproblemsolving.com/1/f/7/1f7c8f86cd0c05915273501b03d7bca9abc30040.png)
and finally ![\[\angle LMC +
\angle LDC = \pi\]](http://latex.artofproblemsolving.com/c/5/6/c564b72cceb119e4a593046876e7809fa5983cfb.png)
and 
intersect the line 
and 
intersect 
at 
at 
and 
respectively. 
and 
and 
.
and 
.Since 
and 
they are obviously similar---1 .Similarly we can prove that triangles 
and 
are similar----2.Now consider the two triangles 
and 
.
=Angl
.
is cyclic .similarly 
is cyclic.Angle 
a
angle 
.Hence triangles 
are similar----3 .Combining 1,2,3 we have Quadrilaterals 
and 
maps point 
and 
are perspective from a line (intersection of 
and 
are concurrent.
,
are concurrent.
