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.
table filled with natural numbers, we say it is a divisor table if:
-th row are exactly all the divisors of some natural number 
,
-th column are exactly all the divisors of some natural number 
,
for every 
.
is given. Determine the smallest natural number 
, divisible by , such that there exists an divisor table, or prove that such does not exist.
be an acute triangle with circumcenter 
. Points 
and 
are chosen on segments 
and 
such that 
. If 
is the midpoint of the arc 
and 
is the midpoint of the arc 
, prove that 
.
, 
, 
be non-negative real numbers, no two of which are zero. Prove that :
that, for all and positive real numbers 
, there is![\[\sum_{i=1}^n \frac{x_i^2}{\sum_{j=1}^i x_j}\geq C \ln n\left(\prod_{i=1}^n x_i\right)^{\frac{1}{n}}\]](http://latex.artofproblemsolving.com/4/8/b/48b9fc4af5a20de7fd9dfc45a74d5aa27e615e46.png)
squares with two game pieces. At the beginning, Alice’s piece is on the first square while Bob’s piece is on the last square. The figure shows the starting position for a strip of 
squares. can Bob ensure a win no matter how Alice plays? can Alice ensure a win no matter how Bob plays?
with incenter 
,
= 
= incenter of 
= incenter of 
= intersection of 
and 
= intersection of 
and 
= midpoint of 
concycle
with circumcenter . 
and 
intersect the altitude from 
to at points and respectively. 
is the circumcenter of triangle 
and 
is the reflection of over . 
is the second intersection of circumcircles of triangles 
and 
. Show that 
are collinear.
, let 
be all positive integers smaller than that are coprime to . Find all 
such that 
for all 
is the largest positive integer that divides both and . Integers and are coprime if 
. such that 
is a perfect square.
be a triangle with incenter . Let be the point of intersection between the incircle and the side , the points and are in the rays 
and 
, respectively, such that 
and 
. Prove that 
.
be real numbers such that 
. Find the value of 
be real numbers such that 
and 
. Find the value of 
with -excenter 
, the foot of the perpendicular from to is . Let the midpoint of segment 
be , 
lies on arc (not containing ) satisfying 
, 
intersects the circumcircle of at 
. If 
and intersect at , the perpendicular bisector of 
intersects 
at 
respectively, prove that 
are concurrent.
such that
in usual Euclidean space such that for every plane 
the intersection 
is finite and nonempty ?
, used in solving Crum's problem of infinitely many neighbourly convex polyhedra (Tietze, Besicovitch). But of course we need each coordinate of odd degree ...
,
as it's cartesian equation (with 
not all zero).
is nonempty because 
always has a real solution (because a polynomial of odd degree always has a real root), and it's finite because 