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.
other knights where friendship is mutual. Prove that King Arthur can place some of his knights around a round table in such a way that every knight is friends with the 
people adjacent to him and between them there are at least 
friendships of knights that are not adjacent to each other.
, call all positive integers that cannot be written as 
(where 
are non-negative integers) bad, and define 
to be the sum of all bad numbers raised to the power of 
. Prove that there exists a positive integer 
, such that for any 
as described, 
divides 
.
be an acute triangle with orthocenter 
, and let 
be a point on the side 
, lying strictly between 
and 
. The points 
and 
are the feet of the altitudes from and , respectively. Denote by 
is the circumcircle of 
, and let 
be the point on such that 
is a diameter of . Analogously, denote by 
the circumcircle of triangle 
, and let 
be the point such that 
is a diameter of . Prove that 
and are collinear.
Prove that
Where 
Prove that
Let 
Prove that
Let 
Prove that
be an integer and 
is a sequence of real numbers for which 
and:
Prove that all elements of the sequence are positive.
is made up of 64 small unit cubes. Exactly 16 of these small cubes must be colored red, subject to the following condition:
, 
, and 
cubes, there must be exactly one red cube.
from the set of positive integers to the set of positive integers such that, for all positive integers 
and 
, there exists a non-degenerate triangle with sides of lengths![\[ a, f(b) \text{ and } f(b + f(a) - 1).\]](http://latex.artofproblemsolving.com/4/a/b/4ab97da27e05d0f9e7762f74d0d0bd2dc8b9842f.png)
,such that the inequality 
holds for all positive reals and for every positive real 
, there exist positive reals 
, such that the equality holds.
, let 
be the real root of equation 
. Prove that
;
.
be an integer, 
a finite set of integers (not necessarily positive) and 
subsets of . Suppose that, for every 
, the sum of the elements of 
is 
. Prove that contains at least 
elements.
ones are written in a circle. Petya and Vasya take turns making 
moves each. In each move, Petya chooses 9 consecutive numbers and decreases each by . Vasya chooses 
consecutive numbers and increases each by 
. They alternate turns, starting with Petya. Prove that Vasya can act in such a way that after each of his moves, there are always at least five positive numbers, regardless of how Petya plays.
ones are written in a circle. Petya and Vasya take turns making moves each. In each move, Petya chooses 9 consecutive numbers and decreases each by . Vasya chooses consecutive numbers and increases each by . They alternate turns, starting with Petya. Prove that Vasya can act in such a way that after each of his moves, there are always at least five positive numbers, regardless of how Petya plays.
.
be the numbers at those special positions.
by 
for some 
,
and 
by 
; changes by 
or 
, and so is the form 
after 
rounds. At least one of those two numbers is positive.
and 
Prove that
and 
Show that
Where 
Where 
and 
,
.
.
