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such that![\[
4k^3 + 4k + 1
\]](http://latex.artofproblemsolving.com/f/3/5/f355e456cd47ecfdb13597d52fec3d848b67c594.png)
is divisible by 
for all positive integers 
.
and satisfying 
.
be a polynomial in one variable with integer coefficients. Prove that the number of pairs 
of positive integers such that 
, is finite.
, let 
. Find all convex sets 
such that![\[\mathcal{E}(B \cup C) = B \cup C.\]](http://latex.artofproblemsolving.com/e/c/3/ec34115c9e6e169c19c739220065f647934a1a5f.png)
and ![$f:[a,b]\rightarrow\mathbb{R}$](http://latex.artofproblemsolving.com/6/7/c/67c320754a36aaec178e61f04bfcbac41c0bf675.png)
be a strictly monotonous function such that 
. Show that 
.
for which there exists a scrictly monotonous function 
such that 
, prove that a convex closed set with a nonempty interior that contains exactly one extreme point must contain a ray. (SOLVED), find all convex closed sets with a nonempty interior that contain exactly one extreme point.
, ∀x∈[0,1].
a distance 
from the start. What is the maximum time it could have taken to travel the distance ?
cells are marked on an 
table so that at least one marked cel is among any four cells of the table which form the figure IMAGE (tbe figure may be rotated). Find the smallest possible value of .
cells are marked on an table so that at least one marked cel is among any four cells of the table which form the figure 
(tbe figure may be rotated). Find the smallest possible value of .
with the following property: cells from a board such that from the remaining cells there are not 4 cells that can be filled with a |-- tile" ( like the tile from your statement)
.
