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.
satisfying
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of square numbers. Prove that: 
is the sum of square numbers
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)
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.
does not have a real root. what is the maximum number of coefficients to be 
?(14 points)
needs have no real roots and 
, it means we need have 
for all real 
. Since we then need 
, it means the number of negative coefficients can be at most 
(out of the total 
coefficients). negative coefficients. Take 
. But 
, while 
. But a polynomial 
has only one real root (root 
), so polynomial 
has no real roots, hence is positively defined. Therefore all terms in the sum above are non-negative, and may be zero only at values 
and/or . However, 
, so 
has no real roots.
has only one real root (root ), so polynomial 
has no real roots.
does not have a real root. what is the maximum number of coefficients to be ?(14 points)
, we see that there doesn't may more than 
, each term is positive.
and 
and 
and add it, we find 
:
and , add them and we find 
