[/quote]
,
respectively 
are marked in such a way that 
and 
for 
.
,
people arranged in a circle, and 
coins are distributed among them, where each person has at least 
coins. Each person is then assigned a random index number in 
such that no two people have the same number. Then every minute, if 
is the number of minutes passed, the person with index number congruent to 
be a positive integer such that 
.
.
be a prime. We say that a sequence of integers 
is a 
,
such that whenever 
,
divides the sum![\[\sum_{k=0}^m (-1)^k {m \choose k} z_k.\]](http://latex.artofproblemsolving.com/1/e/9/1e99c787677d83944ee00e370b7913b1c2728fcd.png)
and 
are 
is a 
board containing all the integers from 
to 
so that each cell contains exactly one number. Two different cells are considered adjacent if they share a common side. Every cell that is adjacent only to cells containing larger numbers is called a valley. An uphill path is a sequence of one or more cells such that:
denote the largest integer such that for any coloring of a 
with two colors, there exists a monochromatic subgraph of 
.
we can find a natural 
,
?
with 
and 
,
.
past 
such that 
.
be on the opposite side of 
from 
such that 
and 
.
intersect the circumcircle of 
again at 
,
intersect the circumcircle of 
again at 
.
,
are collinear.
be a point in the exterior of the circle 
of center 
and radius 
, 
be the tangent segments from 
.Let the line from 
intersect the segment 
, and if the incircle of the triangle 
, then find the area of 
in terms of 
.
be the set of positive integers. Determine all functions 
such that 
is divisible by 
for all positive integers 
.
and 
.
belongs to 
be a positive integer. A graph on 
vertices is given such that the size of the largest clique in the graph is 
if and only if 
and 
are squares of rational numbers and are in Arithmetic Progression
and 
,
,
.
Y by Adventure10, Mango247
Two circles 
and 
intersect at 
and 
. The common tangent of and that is close to than meets at 
and at 
. 
extended meets at 
and 
extended meets at 
. Let 
and 
be the midpoints of 
and 
respectively.
Show that
bisects angle 
.
and 
intersect at 
and 
. The common tangent of and that is close to than meets at 
and at 
. 
extended meets at 
and 
extended meets at 
. Let 
and 
be the midpoints of 
and 
respectively.Show that
bisects angle 
.
,
,
and 
.
concyclic. Similarly, 
concyclic. Then 
.
.
and 
.
.
,
and 
are three similar triangles, and 
are their respective medians. The result follows.
