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)
+1 w
be real numbers such that 
. Find the smallest possible value of 
.
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 
2021 IMOC Problems
-gon 
are colored red and blue in an alternating fashion.
-gon, as do the extensions of the blue sides.
diagonals 
are concurrent.
. What is the least possible positive integer value of such that there exists a non-negative integer for which the set 
is fragrant?
having the following property: for any integers 
whose sum is not divisible by 
, there exists an index 
such that none of the numbers 
is divisible by . Here, we let 
when 
.
and 
Prove that
Let 
Prove that
.
Let 
.
