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 non-negative real numbers, no two of which are zero. Prove that :
that, for all 
and positive real numbers 
, there is![\[\sum_{i=1}^n \frac{x_i^2}{\sum_{j=1}^i x_j}\geq C \ln n\left(\prod_{i=1}^n x_i\right)^{\frac{1}{n}}\]](http://latex.artofproblemsolving.com/4/8/b/48b9fc4af5a20de7fd9dfc45a74d5aa27e615e46.png)
squares with two game pieces. At the beginning, Alice’s piece is on the first square while Bob’s piece is on the last square. The figure shows the starting position for a strip of 
squares. can Bob ensure a win no matter how Alice plays? can Alice ensure a win no matter how Bob plays?
with incenter 
,
= 
= incenter of 
= incenter of 
= intersection of 
and 
= intersection of 
and 
= midpoint of 
concycle
+1 w
be a triangle with incenter . Let be the point of intersection between the incircle and the side , the points and are in the rays 
and 
, respectively, such that 
and 
. Prove that 
.
with 
-excenter 
, the foot of the perpendicular from to is . Let the midpoint of segment 
be , 
lies on arc (not containing ) satisfying 
, 
intersects the circumcircle of at 
. If 
and intersect at 
, the perpendicular bisector of 
intersects 
at 
respectively, prove that 
are concurrent.
, 
, 
, be four points such that no three are collinear and is not the orthocenter of . Let , , 
be the orthocenters of 
, 
, , respectively. Suppose that the lines 
, 
, 
are pairwise distinct and are concurrent. Show that the four points , , , lie on a circle.
and 
be the circumcenter and orthocenter, respectively, of an acute scalene triangle . The perpendicular bisector of 
intersects 
and 
at 
and 
respectively. Let 
denote the intersection of the circumcircles of triangles 
and 
other than .
and 
analogously by repeating this construction two more times. Prove that , , , and are concyclic.
be a quadrilateral inscribed in a circle 
. Let 
, 
, 
, 
, 
, 
, 
, 
. Prove that 
lies over 
.
is circumcircle of an acute triangle . is midpoint of arc 
and is incenter of triangle . Let 
intersect in and for second time in . Let be a point on line 
such that 
is parallel to . Prove that is bisector of angle 
.
with 
and the circle of radius 
centered at 
Let be the midpoint of 
The line 
intersects a second time at 
Let be a point on such that 
Let 
be the intersection of 
and Prove that 
be . A point lies on segment , and lies on segment . Let ray 
. A point , which lies on , bisects and it is on the other side of with respect to . Ray 
, ray 
, and 
. Prove that 
are cyclic.
, then you would submit 1734274).
, then you would submit 1734274).
.
from 
from 
to 
.
is 30-60-90 and 
,
,
and the volume is 
,
.
.
,
.