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 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 acute triangle , let 
be the foot of the altitude from to , and let be the reflection of across . Suppose that the circumcircle of triangle 
intersects line at two distinct points 
and 
. Prove that is the midpoint of 
.
be a quadrilateral inscribed in circle with 
. Let and be the reflections of over lines 
and , respectively, and let be the intersection of lines 
and 
. Suppose that the circumcircle of 
meets at and 
, and the circumcircle of 
meets at and 
. Show that 
.
such that among any 
positive real numbers 
with 
, there exist three that are the side lengths of an acute triangle.
for which 
is a perfect square.
and 
Prove that
Let 
Prove that
