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.
with 
, a tangent to the circumcircle of at point 
intersects the extension of 
at point 
. 
is the midpoint of 
, and 
intersects the circumcircle of at 
. Prove that 
.
be an acute triangle inscribed in a circle 
. Let 
be the orthogonal projection of 
on 
. Let 
be the orthogonal projection of on 
respectively. Let 
be a point such that 
is the diameter of . Prove that the area of the quadrilateral 
is equal to the area of triangle .
be the number of ways in which the set 
can be partitioned into non-empty subsets. Let 
be the number of ways in which the set 
can be partitioned into non-empty subsets such that consecutive numbers belong to distinct subsets. Partitions that differ only in the order of the subsets are considered equal. Prove that 
.
with the set 
such that each element in the permutations has at most one immediate neighbor greater than itself.
be reals such that 
Prove that
Let be reals such that 
Prove that
Let be reals such that 
Prove that
such that there exists 
satisfying 
.
and ![$P_1(x),P_2(x), \ldots P_n(x) \in \mathbb{Z}[x]$](http://latex.artofproblemsolving.com/c/6/4/c64176e52d94032e571b4579e35f9e6adfde675a.png)
such that 
. Prove that exists many 
such that every equation: 
has no real roots
meet at point 
. Points 
and 
lie on 
, 
respectively in such a way that 
and 
. Prove that the line joining the common points of circles 
and 
passes through the mass-center of .
lies inside a triangle on the bisector of angle . Let 
and 
be the circles touching 
and 
at and passing through ; and 
be the common points of and with the circumcircle of distinct from . Prove that the circumcircles of the triangles 
and 
are tangent.
such that for every prime number 
and natural number 
,
is a complete residue system modulo . With 
for every natural number 
and 
.
, consider a prime dividing 
and we see that it's a contradiction.
so 
, and we see that 
.
. If 
, then 
and it's a contradiction.
for all 
.
. I think your solution is still correct tho
. In particular, 
. Now we induct and prove 
.
. Then 
is not a complete residue system 
, so we have a contradiction and 
.
, as the base case is previously verified. works, we win the game. 
and 
Prove that
Where 
![$\mathbb Z[\sqrt n]$](http://latex.artofproblemsolving.com/7/3/8/738c449809946e13bc9a16c316a2233f0a2db4f9.png)
is a UFD.
such that 
.
