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.
with 
. Points 
and 
are on 
and 
respectively such that 
. Lines 
and 
meet at 
. 
is the incenter of 
and 
is the orthocenter of 
. 
and 
are circles with diameter 
and 
, respectively, intersecting each other at points 
and 
. Prove that and lie on 
.
. Prajit then tries to place n points such that no four points are concyclic and no 
points are collinear in Euclidean plane, such that Shining cannot find a group of three points such that their circumcircle contains none of the other remaining points. Is he always able to do so?
be an integer, and let 
be positive real numbers such that 
. Prove that![\[(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.\]](http://latex.artofproblemsolving.com/7/7/f/77fb719e975225b59c64f1f3aa102c98a6c25f50.png)
not be a trapezoid such that there is a circle centered at that is tangent to the four sides 
, 
, 
, and 
. Let , , 
, and 
be the circumcenters of the triangles 
, 
, 
, and 
, respectively. Prove that there is a circle containing the circumcenters of the triangles 
, 
, 
, and 
.
such that
satisfies 
. Find the value of 
.
with 
, a tangent to the circumcircle of at point 
intersects the extension of 
at point 
. 
is the midpoint of 
, and 
intersects the circumcircle of at 
. Prove that 
.
be a parallelogram of center 
. Prove that for any point 
, there exist unique points 
and 
such that is the center of mass of 
.
plane let 
(with 
) define parallelogram whose center is 
and sides are defined by the equations: 
. Taking the points 
, 
, and 
the centroid of 
is computed per:
(i);
(ii).
, then:
(iii);
(iv).
unique points 
for any 
(for 
) on side .
is the centroid of 
if and only if 
passes through the midpoint of 
and 
passes through the midpoint of 
, if and only if passes through the midpoint of and passes through the midpoint of , if and only if 
is on the image of by the homothety of centre 
and ratio 
and 
is on the image of by the homothety of centre and ratio . Hence the unique points 
and 
such that is the centroid of are the intersections of the images of and by the homothety with centre and ratio with and , respectively.
is the reflection of wrt then 
, 
and 
.
. Then 
and 
, so we may write 
, 
and 
, for 
. Then is the centroid of if and only if 
, if and only if 
. Since 
and 
are linearly independent, this happens if and only if 
and 
. In this case 
and 
, so both and lie on the segments and not on their extensions. on the line there exist unique points on the line 
and on the line 
such that is the centroid of (and both tell us exactly where on the line they are, that is, on what ray they are, for example, not only how to construct them).
such that 
.
and 
Prove that
Where 
)
and 
Prove that
![$$a+\dfrac{2}{a}+\dfrac{3}{b}\geq 4\sqrt[4]{\frac23}$$](http://latex.artofproblemsolving.com/b/c/1/bc1ceb6a41d5fc0598cdd0cc455875087f9d4585.png)
![$$\dfrac{2}{a}+\dfrac{3}{b}\geq 2\sqrt[4]3$$](http://latex.artofproblemsolving.com/f/2/2/f22009d039177d6343c55023754310e0e59c1744.png)
![$$3a+\dfrac{2}{a}+\dfrac{3}{b}\geq \sqrt[4]{354+66\sqrt{33}}$$](http://latex.artofproblemsolving.com/e/c/9/ec9133ad1679ba58cefdb5de88de5550e6903d83.png)
