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 
. Prove that
Let 
. Prove that
Let 
. Prove that
Let . Prove that
and 
. Then we should find this ratio: 
.![\[S_{EGC}=(a+x)^2+\frac{x^2}{2}-\frac{(a+x)(a+2x)}{2}-\frac{a(a+x)}{2}=\frac{x^2}{2}.\]](http://latex.artofproblemsolving.com/2/a/7/2a7271b3023b31cbdadac9161187b92f1a292986.png)
![\[S_{DEFBC}=(a+x)^2+x^2+\frac{ax}{2}.\]](http://latex.artofproblemsolving.com/c/f/0/cf022939edf4bba3c1ce126b096a1e193665deee.png)
![\[S_{DEFBC}=16S_{EGC}\Rightarrow a^2+2.5ax-6x^2=0\]](http://latex.artofproblemsolving.com/e/a/6/ea6e9085f8aac29b7c60539cc726c5be3c33a48e.png)
![\[\left(\frac{a}{x}\right)^2+2.5\frac{a}{x}-6=0,\quad \frac{a}{x}>0\Rightarrow\frac{a}{x}=\frac{3}{2}.\]](http://latex.artofproblemsolving.com/f/3/0/f3047f7e354ad1e790923424d1205422532724df.png)
Therefore,![\[\left(\dfrac{x}{a+x}\right)^2=\left(\dfrac{1}{\frac{a}{x}+1}\right)^2=\frac{4}{25}.\]](http://latex.artofproblemsolving.com/9/3/8/9388ead9972342deafa8461cdcb8961e91254ed0.png)
Hence answer is: 
.
.
for which 
is a perfect square.
and 
Prove that
Let 
Prove that
lies on side 
is drawn outwards 
is 
of the area of pentagon 
.
