AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
be a prime. We say that a sequence of integers 
is a -pod if for each 
, there is an 
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 -pods, then the sequence 
is a -pod.
be a positive integer. A Nordic square is an 
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:, the smallest possible total number of uphill paths in a Nordic square., let 
denote the largest integer such that for any coloring of a 
with two colors, there exists a monochromatic subgraph of isomorphic to 
. Is it true that for each positive integer 
we can find a natural 
such that for any integer 
, 
?
with 
and 
, let 
be the midpoint of 
. Choose point 
on the extension of 
past 
and point 
on segment 
such that lies on 
. Let 
be on the opposite side of 
from 
such that 
and 
. Let 
intersect the circumcircle of 
again at 
, and let 
intersect the circumcircle of 
again at 
. Prove that , 
, and 
are collinear.
be a point in the exterior of the circle 
of center 
and radius 
, and let 
, 
be the tangent segments from to the circle. On the segment consider the point 
such that 
.Let the line from parallel to 
intersect the segment at . If is a point on the segment other than so that 
, and if the incircle of the triangle has radius 
, 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 
.
be a set of integers containing the numbers 
and 
. Suppose further that any integer root of any non-zero polynomial with coefficients in also belongs to . Prove that 
belongs to .
be a positive integer. A graph on 
vertices is given such that the size of the largest clique in the graph is . Prove that there exists a vertex that is present in every clique of size 
if and only if 
and 
are squares of rational numbers and are in Arithmetic Progression is an integer.
such that, for all positive integers 
and 
, if divides 
, then divides 
.
+1 w
