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.
, such that 
and 
.
such that 
is prime, 
with 
and ?
be a function such that for every regular 
-gon 
we have 
. Prove that 
for all reals 
.
+1 w
be a triangle (
). 
are the altitudes of the triangle. The bisector of 
intersects 
at 
. Let 
be a point such that 
and 
. Prove that 
passes through the midpoint of 
.
with 
and at most one of them being zero, show that![\[\frac{1}{\max(a^2,b)}+\frac{1}{\max(b^2,c)}+\frac{1}{\max(c^2,a)}\geq\frac{27}{4}\]](http://latex.artofproblemsolving.com/9/e/1/9e14a9ef2d1a0a98006bcf716fa33c3d9ea29bd5.png)
be an integer. Find all sequences 
satisfying the following conditions:![\[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 + n;
\]](http://latex.artofproblemsolving.com/3/c/5/3c509ec2e9013e8d3be492c8eb44a7c33841b74e.png)
![\[ \text{ (b) } a_{i + 1} + a_{i + 2} + \ldots + a_{i + n} < a_{i + n + 1} + a_{i + n + 2} + \ldots + a_{i + 2n} \text{ for all } 0 \leq i \leq n^2 - n.
\]](http://latex.artofproblemsolving.com/9/7/d/97d2a467d1c0dc8594ec024c3bb9b8c87ee85b19.png)
and 
. Prove that
Equality holds when 
or 
and 
, Ivica and Marica play the following game: Given two piles of cookies, on each turn a player takes 
cookies from one of the piles, of which he eats 
and puts of is arbitrary in every move. Players take turns alternatively, with Ivica going
for which Marica has a winning strategy.
be non-negative real numbers such that 
Prove that
be the circumcenter, symmedian point of 
. Show that the tangent of 
at 
,the tangent of 
at 
, the tangent of 
at 
are concurrent.
be distinct natural numbers each of which has exactly 
positive integer factors. For instance, 
has exactly positive integer factors 
. Assume that no prime larger than 
divides any of the 
's. Show that there must be some perfect cube among the 's.
and denoting 
the coordinates of 
the areas of 
are given by![$\frac{[PAB]}{[ABC]}=\left |\frac{z}{x+y+z} \right| \ , \ \frac{[PBC]}{[ABC]}=\left |\frac{x}{x+y+z} \right|$](http://latex.artofproblemsolving.com/e/1/f/e1fb807e09001b7baaa09e7cec821f650770901d.png)
![$\frac{[PCD]}{[ABC]}=\left |\frac{qx-py}{(x+y+z)(p+q+r)} \right| \ , \ \frac{[PDA]}{[ABC]}=\left |\frac{ry-qz}{(x+y+z)(p+q+r)} \right|$](http://latex.artofproblemsolving.com/6/d/6/6d636f4a5dde474726e71d6ac1e25c965a1f5caa.png)
![$[PAB] \cdot [PCD]=[PBC] \cdot [PDA] \Longrightarrow z|qx-py|=x|ry-qz| \Longrightarrow$](http://latex.artofproblemsolving.com/f/9/f/f9f0b0cda72aa265d14612d95508ce30a2c4a2b5.png)
or 
is on a circumconic 
of through 
is the isogonal of the line 
WRT where 
is the isogonal conjugate of 
WRT and 
is on 
such that 
, 
, 
, 
math=inline]$ CPD $[/mathmath=inline]$ DPA $[/math 
is dioganals of the quadriteral 
.
& 
.
,![\[S_{PAB}\cdot S_{PCD} = S_{PBC}\cdot S_{PDA}\]](http://latex.artofproblemsolving.com/c/9/0/c90fbbc074bc27703dec46d97cd9686811666293.png)
denotes the area of triangle 
)
