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 real numbers such that ![\[ab^2+bc^2+ca^2=6\sqrt{3}+ac^2+cb^2+ba^2.\]](http://latex.artofproblemsolving.com/3/d/0/3d015292c249a69a44d5f923092201d7a271b896.png)
Find the smallest possible value of 
.
distinct positive integers and then writes in blue the 
sums of each two red numbers. At most how many of the blue numbers can be prime?
, let a line 
pass through the orthocenter and not through point 
. The line intersects line 
at 
. A line passing through and perpendicular to meets the circumcircle of triangle at 
. Let the feet of the perpendiculars from 
to be 
, respectively. Define line 
as the line passing through 
and perpendicular to , and line 
as the line passing through 
and perpendicular to 
. Prove that if 
is the reflection of the intersection of and across , then 
.
, let 
be the number of positive integer divisors of .
we have that:![\[ d(a)+d(b) \le d(\gcd(a,b))+d(\text{lcm}(a,b)) \]](http://latex.artofproblemsolving.com/a/3/b/a3bfe462d3888517939eec8a2827ee6bd8d627a1.png)
and determine all pairs of positive integers where we have equality case.
. A convex polygon is said to be -good if it contains lattice points where any three of them are not collinear.-good convex polygon with area at most 
.
so that any -good convex polygon has area at least 
.
and 
be concentric circles, with in the interior of . From a point on one draws the tangent 
to (
). Let 
be the second point of intersection of and , and let 
be the midpoint of . A line passing through intersects at 
and 
in such a way that the perpendicular bisectors of 
and 
intersect at a point 
on . Find, with proof, the ratio 
.
with 
and let be 
it’s circumcircle. Diameters 
and 
are drawn. Circle 
interescts 
at 
. Circle 
intersects 
at 
.(
lies between 
and ). Prove that lines 
and 
intersect at circle 
.
has been partitioned into disjoint pairs 
(
) so that for all 
, 
equals 
or 
. Prove that the sum ![\[ |a_1-b_1|+|a_2-b_2|+\cdots +|a_{999}-b_{999}| \]](http://latex.artofproblemsolving.com/0/4/4/04422a7d2266b92a6b3f215b698f078cda85091f.png)
ends in the digit 
.
where 
is a parallelogram.
. is connected to the midpoint of ![$[CT]$](http://latex.artofproblemsolving.com/f/3/f/f3f8e9913dc696e543e6a34d169ec3deb276c84d.png)
, point 
to the midpoint of ![$[DT]$](http://latex.artofproblemsolving.com/6/d/d/6dd49550ab42ec2f047eaf4b357ac4e7d1387b1e.png)
, to the midpoint of ![$[AT]$](http://latex.artofproblemsolving.com/6/e/3/6e397da3c2a38b457f7b3504c5cb2a488a64f3a6.png)
and point to the midpoint of ![$[BT]$](http://latex.artofproblemsolving.com/6/a/4/6a4b1bcea7921e762a359321ab861b6937f04782.png)
.
.
.
![\[ g_0 = \dfrac{1}{3} \left( \dfrac{m + a + b}{3} + \dfrac{m + b + c}{3} + \dfrac{m + c + a}{3} \right) = \dfrac{3m + 2a + 2b + 2c}{9} = \dfrac{1}{3} m + \dfrac{2}{3} \left( \dfrac{a + b + c}{3} \right). \]](http://latex.artofproblemsolving.com/6/9/1/691f6cb3a4765645e954b0d5c20331c1a2e19e09.png)
Let 
be the centroid of 
, the above implies 
so 
passes through which is fixed.
as 
Prove that 
is prime for all 
.
