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.
order, and if the constituents of its first, second, ..., rows are the first 
figurate numbers of the first, second, ..., orders respectively, show that it's value is 
, 
be the orthocenter of it and 
be any point on the side 
. The points 
are on the segments 
, respectively, such that the points 
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
. Prove that the points 
and lie on the same line.
positive such that 
, then there exists 
positive such that 
converges and 
diverges ?
is continuous, positive with 
, does 
continuous and positive exist on 
such that 
converges and 
diverges ?
is in the 
class, increasing : I chose 
. With this idea, I had the idea to define 
but it is not clear that it is ok, even if is increasing. positive such that , then there exists positive such that converges and diverges ? And if is increasing ? positive increasing such that 
, then 
diverges ? positive increasing such that , then 
converges ? is positive increasing and such that 
does not converge to 
, can diverge ? only to be continuous ?
and 
are positive integers, we say that almost divides if divides at least one of 
and 
. We call a positive integer almost prime if the following holds: for any positive integers 
such that almost divides 
, we have that almost divides at least one of and . Determine all almost prime numbers.
can be expressed as a polynomial in 
for any 
.
and 
are polynomials in .
be expressed as a polynomial in with real coefficients? That is, ![$$ \exists Q \in \mathbb{R} [X] , \;\;\; \text{s.t.} \;\;\; \forall t \in \mathbb{R} , \;\; \cos ( \sqrt{2}\, t ) = Q( \cos t ) \;\;?$$](http://latex.artofproblemsolving.com/e/d/d/eddda470a20e23492ba00f9460bbf124a75e11c5.png)
Incidentally, this question came to my mind while reading the following thread: https://artofproblemsolving.com/community/c51293h3562250
holds inequality 
and 
, then find the minimum value of ![$\sqrt{ad^3 g^4} +\sqrt[3]{be^3 h^4} + \sqrt[8]{cf^3 k^4}$](http://latex.artofproblemsolving.com/b/f/a/bfa17e8bb4cc05a84ed6a757f506d6d0ed2b0081.png)
is an odd number, let 
be a positive integer. If 
then 
such that 
. Find the minimum: 
points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least 
scalene triangles with vertices of that color.
and 
.
?
be positive integers satisfying ![\[\frac{ab}{a+b}+\frac{cd}{c+d}=\frac{(a+b)(c+d)}{a+b+c+d}.\]](http://latex.artofproblemsolving.com/b/8/c/b8c3fe14640a2dcb30b755b2347721da5d080a78.png)
Determine all possible values of 
.
users, some pairs of whom are friends. Whenever user 
is friends with user 
, user is also friends with user . Events of the following kind may happen repeatedly, one at a time:, , and 
such that is friends with both and , but and are not friends, change their friendship statuses such that and are now friends, but is no longer friends with , and no longer friends with . All other friendship statuses are unchanged.
users have 
friends each, and users have friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user.
as an example. Each eigenvalue 
of 
is a root of 
which is irreducible by Eisenstein. And also because it has no rational root. Also ![$[Q(t):Q]\le 2$](http://latex.artofproblemsolving.com/e/0/a/e0a35fdece3a667234bf58b762cd33c123b6b812.png)
since 
has degree 2. Contradiction. Hence for we have no solution in 
. The equation is solvable iff 
is the cube of an integer.
where 
Hence 
, and consequently also , is the cube of an integer. not divisible by 
., however, the equation is always solvable.
doesn't divide , then 
and 
.
If 
then 
and 
, where 
.
so that is an integer matrix ? (In general, the integer matrices are not sufficient). If 
then is diagonalizable over 
and 
, where 
.
. so that is an integer matrix ? divides and 
, then 
, is diagonalizable over and 
. is not a cube, then 
is irreducible over 
and we can group the eigenvalues by , that is, 
.
, where 
. so that is an integer matrix ?
is solvable over 
for every 
.
,
.