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.
be an integer, and let 
be positive integers. Show that there exist positive integers 
satisfying the following three conditions:
for 
the remainders of on division by 
are pairwise different; and
denotes the integer part of real number 
, that is, the largest integer that does not exceed .)
, 
, 
be positive real numbers with 
. Show that
Where 
with circumcenter 
and orthocenter 
. Line 
meets 
at 
respectively.
as the circumcenter of 
. The circumcircle of meets the circumcircle of 
again at 
, 
. Line meets circumcircle of 
again at 
, 
and circumcircle of meets circumcircle of again at 
, 
. Define 
as the intersection of 
and and 
meets circumcircle of at points 
.
and circumcircle of are tangent to each other.
add up to zero. Find the maximum of 
in term of 
's, when 
's vary in real numbers such that 
., , be positive real numbers with . Show that ![\[ a^2 + b^2 + c^2 \geq \sqrt 3 abc. \]](http://latex.artofproblemsolving.com/6/c/7/6c779e29d3ae8516bae06fe2e9a7414869940027.png)
be real numbers such that 
. Find the smallest possible value of 
.
+1 w
grid has 
cells painted such that the following condition holds:.
and it's circumcircle 
. With center 
, we construct a circle 
that intersects arc 
of circle , that doesn't contain 
, at point and arc 
, that doesn't contain 
, at point 
. Suppose that the intersection point of lines 
and 
lies on circle . Prove that line 
is perpendicular on line 
.
, and for all 
, define the sequence 
by the recurrence![\[
a_{n+1} = a_n^2 + 1
\]](http://latex.artofproblemsolving.com/3/1/1/3114595ded8c8961d611afaab9ad19e254973001.png)
Prove that there is no natural number 
such that![\[
\prod_{k=1}^{n} \left( a_k^2 + a_k + 1 \right)
\]](http://latex.artofproblemsolving.com/a/4/f/a4f88ee7e3d4e53e369319f74cd9b12be26c9681.png)
is a perfect square.
be distinct real numbers such that![\[
a + b + c + \frac{1}{abc} = \frac{19}{2}
\]](http://latex.artofproblemsolving.com/5/1/e/51eef77358db2b6c98f0fb0fb4e30db837a03642.png)
Find the maximum possible value of 
.
be a real numbers such that 
Prove that
Where 
with 
be the intersection of and 
Let 
meet again 
at 
Called be the midpoint of Assume that 
Prove that is the tangent of and is the tangent of 
be the orthocenter of an acute-angled triangle and a point on the circumcenter of triangle . Prove that the Simson line of bisects the segment ![$[P H]$](http://latex.artofproblemsolving.com/5/8/7/587225c92de7ccb1a3837835001032884cf125b8.png)
.
for which 
is a perfect square.
and 
Prove that
Let 
Prove that
