[/quote]
,
satisfying
such that for all real 
:
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of 
is the sum of 
be positive reals with 
.![\[\sum_{i=1}^n \frac{x_i}{2-x_i} \ge \frac{n}{2n-1}\]](http://latex.artofproblemsolving.com/f/6/5/f65f6fce4465125dfacf002f0659c898df7ae150.png)
and 
are the incenter and the 
,
be the midpoint of arc 
on the circumcircle, and 
be the foot of the
.
intersects the circumcircle again at 
. For
on the circumcircle of triangle 
intersect 
. Prove
are concyclic.
squares, we cover the area with several S-Tetrominos (=Z-Tetrominos) along the square so that in every square, there are two or fewer tiles covering that (tiles can be overlap). Find the maximum possible number of squares covered by at least one tile.
satisfies
for any non-negative integers 
and 
.
.
be a function from the Euclidean plane to the real numbers such that 
for any acute triangle 
with circumcenter 
,
and orthocenter 
.
be the positive real number. Prove that
,
grid. Alice goes first, and they take turns drawing a black point from the coordinate set![\[\{(i, j) \mid i, j \in \mathbb{N}, 1 \leq i, j \leq n\}\]](http://latex.artofproblemsolving.com/4/3/f/43f7493d1dff43079eb5481ee17bc8f16d7b9553.png)
There is a constraint that the distance between any two black points cannot be an integer. The player who cannot draw a black point loses. Find all integers 
Y by Adventure10, Mango247
Let 
, 
, 
, 
, 
, 
be six points lying on the periphery of a circle (in this order).
Let the chords
and 
meet at a point 
; let the chords and 
meet at a point 
; let the chords and meet at a point 
.
Prove that
.
Comment on the source. The problem is inspired by Stergiu's proof in http://www.mathlinks.ro/Forum/viewtopic.php?t=50262 post #5.
Darij
, 
, 
, 
, 
, 
be six points lying on the periphery of a circle (in this order).Let the chords
and 
meet at a point 
; let the chords and 
meet at a point 
; let the chords and meet at a point 
.Prove that
.Comment on the source. The problem is inspired by Stergiu's proof in http://www.mathlinks.ro/Forum/viewtopic.php?t=50262 post #5.
Darij
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