[/quote]
,
has circumcircle 
and circumcenter 
.
with center 
intersects the segment 
at points 
and 
,
,
are all different and lie on line 
and 
be the points of intersection of 
be the second point of intersection of the circumcircle of triangle 
and the segment 
.
be the second point of intersection of the circumcircle of triangle 
and the segment 
.
and 
are different and intersect at the point 
.
.
of positive integers such that 
is an integer.
be an acute triangle, 
its orthocenter and 
be a point on the parallel through 
and 
a point on the parallel through 
such that 
and 
and 
are the reflections of 
and 
are the intersections of 
and 
respectively with the circumcircle of 
and 
lies on 
.
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 
meet 
at 
is the midpoint of arc 
holds, then what is the value of 
?
and 
and 
is a prime number
.
as points on the parallels through 
are positioned around the sides of 
,
and 
,
is the reflection of 
,
and the parallel through 
,
and the perpendicular through 
and 
a point on 
,
.
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 
at 
and 
for the second time, respectively. Let 
and 
be circumcenters of triangle 
and 
,
are collinear.
,
are also in 
for positive integers 
Y by khan.academy, RANDOM__USER
Let 
and 
be positive integers. In an 
grid, two cells are considered neighboring if they share a common edge. Kritesh performs the following actions:
1. He begins by writing
in any cell of the grid.
2. He then fills each remaining cell with a non-negative integer such that the absolute difference between the numbers in any two neighboring cells is exactly
.
Kritesh aims to fill the grid in a way that maximizes the sum of the numbers written in all the cells. Determine the maximum possible sum that Kritesh can achieve in terms of and .
(Kritesh Dhakal, Nepal)
and 
be positive integers. In an 
grid, two cells are considered neighboring if they share a common edge. Kritesh performs the following actions:1. He begins by writing
in any cell of the grid.2. He then fills each remaining cell with a non-negative integer such that the absolute difference between the numbers in any two neighboring cells is exactly
.Kritesh aims to fill the grid in a way that maximizes the sum of the numbers written in all the cells. Determine the maximum possible sum that Kritesh can achieve in terms of
and .(Kritesh Dhakal, Nepal)
.
is 
since there is a path between 
.
for fixed 
.
where 
constant 
maximum occurs when 
.
and this maximum value is equal to 
