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.
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
with 
. Points 
and 
are on 
and 
respectively such that 
. Lines 
and 
meet at 
. 
is the incenter of 
and 
is the orthocenter of 
. 
and 
are circles with diameter 
and 
, respectively, intersecting each other at points 
and 
. Prove that and lie on 
.
. Prajit then tries to place n points such that no four points are concyclic and no 
points are collinear in Euclidean plane, such that Shining cannot find a group of three points such that their circumcircle contains none of the other remaining points. Is he always able to do so?
be an integer, and let 
be positive real numbers such that 
. Prove that![\[(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.\]](http://latex.artofproblemsolving.com/7/7/f/77fb719e975225b59c64f1f3aa102c98a6c25f50.png)
not be a trapezoid such that there is a circle centered at that is tangent to the four sides 
, 
, 
, and 
. Let , , 
, and 
be the circumcenters of the triangles 
, 
, 
, and 
, respectively. Prove that there is a circle containing the circumcenters of the triangles 
, 
, 
, and 
.
such that
satisfies 
. Find the value of 
.
for all 
and 
for all 
for some integer 
and , .
dividing 
for some integer . Then, the numbers 
wouldn't leave distinct remainders modulo . Therefore, 
for all integers .
, we can't have 
so we must have 
. Similarly, if 
, we can't have 
so we must have 
and we are done. 
.
, , and for 
and for 
.
otherwise then we could chose a such that 
, giving a contradiction. Notice that the numbers 
and 
both leave distinct remainders 
so we must have 
. Now assume we have that 
changes such that and then plugging in 
gives 
, a contradiction.
and 
Prove that
Where 
Where 
such that 
.
such that the numbers![\[n, f(n),f(f(n)),\dots,f^{m-1}(n)\]](http://latex.artofproblemsolving.com/b/3/0/b303b6dd25b8e172f41737ffa6c136dcc4db2266.png)
are distinct modulo 
for all integers 
with 
is defined by 
and 
for 
.
