AoPS Academy
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.
has at most one solution continuous for 
(
),
is a fixed real number. (A. 7)
converge?
, define
where 
is the identity matrix. If for some matrix that 
is identity, does it follow that is zero?
be real numbers. Find the ranges of such that the improper integral 
converges.
are group. If 
, prove that 
or 
. 
are complex numbers such that
and if 
denotes the sum of all products of these numbers taken 
at a time, prove that
whenever the denominator of the left-hand side is different from 
.
for any positive reals satifying 
?
are real number satisfying
show that the equation 
has at least one real root.
on the board. For each procces, the stutent deletes the number x and replaces it with either 
or 
or 
. Is this possible to make the number 
on the board?
on the board. For each procces, the stutent deletes the number x and replaces it with either or or . Is this possible to make the number on the board?
in mod7.Proof is easy,just take 
a
.
.