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 positive reals with 
. Then show that![\[\sum_{i=1}^n \frac{x_i}{2-x_i} \ge \frac{n}{2n-1}\]](http://latex.artofproblemsolving.com/f/6/5/f65f6fce4465125dfacf002f0659c898df7ae150.png)
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of square numbers. Prove that: 
is the sum of square numbers
+1 w
and 
are the incenter and the 
-excenter of triangle 
, respectively.
be the midpoint of arc 
on the circumcircle, and 
be the foot of the to 
. The line 
intersects the circumcircle again at 
. For
on the circumcircle of triangle , let 
intersect at 
. 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 
. Determine the number of possible sets of integers 
.
be a function from the Euclidean plane to the real numbers such that 
for any acute triangle 
with circumcenter 
, centroid 
and orthocenter 
. Prove that 
is constant.
be the positive real number. Prove that
, such that for all positive reals 
, the following is true:
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 such that Alice has a winning strategy. grid, where 
. Prove that the remainder can be covered by copies of the figures of or 
unit squares depicted in the drawing below. for which there exists a positive integer 
with the property that is divisible by and 
is divisible by 
.
denote the set of real numbers. Find all functions 
such that![\[f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y)\]](http://latex.artofproblemsolving.com/5/b/e/5be34cb26d276bae3c60fb299ad66d83bc158a01.png)
.
be a triangle with 
. The point is the midpoint of the side . A circle with center at vertex intersects the line at points and . Prove that 
.
and 
Since 
, thus
We want to show that 
., 
and 
over angle bisector 
and obviously we see that goes to and let point 's reflection point be 
. Let be the intersection point of angle bisector and line .
, hence we only need to show that 
.
and since we reflected over the line 
, then 
and 
.
Hence, we are done yay.
: symmetrical point of C to bisector of 
