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.
of positive integers is called central if for every positive integer 
, the arithmetic mean of the first 
terms of the sequence is equal to .
, 
, 
, 
of positive integers such that for every central sequence 
, 
, 
, , there are infinitely many positive integers with 
.
+3 w
be an acute triangle. Points 
, and 
lie on a line in this order and satisfy 
. Let 
and 
be the midpoints of 
and 
, respectively. Suppose triangle 
is acute, and let 
be its orthocentre. Points 
and 
lie on lines 
and 
, respectively, such that 
and are concyclic and pairwise different, and 
and are concyclic and pairwise different. Prove that 
and are concyclic.
+2 w, let 
be all the positive integers smaller than that are coprime to . Find all 
such that
for all 
.
be non-negative real variables with 
Prove that
Equality holds iff 
or 
where 
+1 w
be a convex quadrilateral such that diagonals 
and 
intersect at right angles, and let 
be their intersection. Prove that the reflections of across 
are concyclic.
we are given :
and one side length of the triangle equals 
, then under these conditions find the maximum area of the triangle .
be a trapezoid with parallel sides 
and 
, where 
. A circle passing through and 
intersects 
again at 
respectively. Prove that 
are concurrent.
in it. Prove that there are an infinite number of perfect cubes in this progression. (A perfect cube is an integer of the form 
, where 
is an integer. For example, 
, 
and are perfect cubes.)
there are elements in an arithmetic progression in the order 
such that 
.
days later (but perhaps earlier, as well). Hedvig received the trophy today. Determine all integers 
for which we can be absolutely certain that she cannot receive the trophy again in days, given the above information.
. Prove that if the leading coefficient of the polynomial equals then the greatest common divisor of the other coefficients also equals .
. Prove that if the leading coefficient of the polynomial equals then the greatest common divisor of the other coefficients also equals .
. Prove that if the leading coefficient of the polynomial equals then the greatest common divisor of the other coefficients also equals .
where 
and let 
be any prime dividing all 
any integer root : 
implies 
and so 
. Motivation is observing why 1 can't be a root.
?
and 
Prove that
Let 
Prove that
and 
Prove that
