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.
and integers 
, prove that for any 
positive numbers 
satisfying 
for 
(with the convention 
), the following inequality holds:![\[
\sum_{j=1}^n a_j^{2m-1} \leq \frac{mt}{2} \left( \sum_{j=1}^n a_j^{m-1} \right)^2.
\]](http://latex.artofproblemsolving.com/5/c/d/5cd52e9ce307563bde96f697a75079229d4bfcc2.png)
is inscribed in circle 
. The interior angle bisector of angle 
intersects side 
and at 
and 
(other than ), respectively. Let 
be the midpoint of side . The circumcircle of triangle 
intersects sides 
and 
again at 
and 
(other than ), respectively. Let 
be the midpoint of segment 
, and let 
be the foot of the perpendicular from to line 
. Prove that line 
is tangent to the circumcircle of triangle 
.
prove that
be an acute angled triangle with 
and incircle 
. Let touch the sides 
and at 
and 
respectively. Let 
and 
be points outside 
satisfying ![\[\angle BDX = \angle XEA = \angle YDC = \angle AFY = 45^{\circ}.\]](http://latex.artofproblemsolving.com/4/7/0/470b0486fa1342c4b6444ec1e3731c23bbb842fb.png)
Prove that the circumcircles of 
and meet at a point 
.
be a fixed integer. The cells of an 
table are filled with the integers from 
to 
with each number appearing exactly once. Let be the number of unordered quadruples of cells on this board which form an axis-aligned rectangle, with the two smaller integers being on opposite vertices of this rectangle. Find the largest possible value of .
of 
touch at , 
, let 
denote the line tangent to through 
. Define 
such that 
. Prove that the circumcenter 
of lies on 
.
and walks randomly. The probability of any point reaching all adjacent points is equal. Find the probability of the ant reaching without passing through 
with 
prime and ![\[a^p+b^p=p!.\]](http://latex.artofproblemsolving.com/f/e/b/feb27894b1f6497559eeb094cfe92256bc26688e.png)
be the incenter of triangle . Let 
, 
, and 
be the orthocenters of triangles 
, 
, and 
, respectively. Prove that the lines through , , and , parallel to 
, 
, and 
, respectively, are concurrent.
be triangle, inscribed in parabola. Tangents in points 
forms triangle . Prove that 
. (
is area of triangle 
).
be triangle, inscribed in parabola. Tangents in points forms triangle . Prove that . ( is area of triangle ).
be any point on the Steiner circumellipse of 
and let 
be the cevian triangle of WRT 
Then ![$[DEF]=2 \cdot [ABC].$](http://latex.artofproblemsolving.com/3/3/e/33ec9a123e2b3b63c128a4fe7fe4aa12551261a1.png)
be the antimedial triangle of (
against 
). The Steiner circumellipse 
of is obviously the Steiner inellipse of tangent to its sides at 
are collinear on the polar of WRT and similarly 
and 
are respectively collinear. Thus by Pappus theorem for the hexagon 
it follows that 
is at infinity, i.e. 
and similarly we get 
![$[DEF]=[DYZ]=[CYZ]=2\cdot [ABC]. \ \blacksquare$](http://latex.artofproblemsolving.com/b/4/0/b4028b012dc90ae29c0bc707cbf447409f6ab058.png)
be the perspector of the parabola. Since the center of any inconic is the isotomcomplement of its perspector (in the parabola case at infinity), then it follows then that the isotomic conjugate of WRT must be on the line at infinity is on the Steiner circumellipse of and the result follows from the lemma.
the equation 
such that 
and 
.![\[\sum_{i=1}^n (a_i-a_{i+1})^3\leq \frac{1}{2\sqrt{3}} (\sum_{i=1}^n a_i )^3,\]](http://latex.artofproblemsolving.com/8/1/0/810f07e3394370ce06aec1e7025bcf5cc5433d60.png)
where 
.
