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 
be real numbers. Let 
be the matrix with entries ![\[a_{ij} = \begin{cases}1,&\text{if }x_i + y_j\geq 0;\\0,&\text{if }x_i + y_j < 0.\end{cases}\]](http://latex.artofproblemsolving.com/c/0/8/c0856fe7b2b113180659952c4d216a01d8849f64.png)
Suppose that 
is an 
matrix with entries 
, 
such that the sum of the elements in each row and each column of is equal to the corresponding sum for the matrix 
. Prove that 
.
be an acute triangle and let 
be an interior point for it such that 
. Prove that
When does equality hold?
be positive reals. Prove that there is a cyclic permutation 
of 
such that the inequality:![\[
\frac{a}{xa + yb + zc} + \frac{b}{xb + yc + za} + \frac{c}{xc + ya + zb} \geq \frac{3}{x + y + z}
\]](http://latex.artofproblemsolving.com/5/5/3/553527fc40a1b7d29dbdfb4b5e63635308f46c85.png)
and 
.
be real numbers such that 
.![\[
3 + (2 - \sqrt{3}) \cdot \frac{(b-c)^2}{b+(\sqrt{3}-1)c} \leq a+b+c
\]](http://latex.artofproblemsolving.com/5/a/8/5a83eaa7470bcca6546599ff4e296e4efec69b16.png)
and determine all the cases when the equality occurs.
+1 w
have the same incenter 
.Prove that 
is concyclic iff 
is concurrent
be the nonnegative integers in 
where 
. Define 
. Prove that 
.
such that 
where 
, and the statement is still true when 
? such that 
where 
, and the statement is still true when ?.
be a scalene triangle with circumcircle 
and circumcircle 
. Denote 
and 
as the midpoints of 
and 
, respectively. The altitude from 
to intersects again at 
. The line 
meets a second time at 
, and 
intersects at 
. The perpendicular bisector of intersects 
at 
and 
at 
. Let the circumcircle of 
intersect again at 
, and let the line 
intersect at 
.
is the reflection of over , prove that the intersection of 
and 
lies on the circumcircle of 
.
be real numbers such that 
and 
and determine the equality cases.
be positive real integer such that 
Prove that
such that 
and 
is tetrahedron 
, 
, 
, 
, 
,
then![$$\sum_{1\le i <j \le 6}(a_i + a_j)^2 \ge 4\sqrt3 S[ABCD]$$](http://latex.artofproblemsolving.com/c/c/1/cc193bab1ea1e3eabbb6a998c4cb07f7c0742d6f.png)
![$S[ABCD]$](http://latex.artofproblemsolving.com/5/b/3/5b38740672d504935d8d03b733de99219c94579a.png)
is total area of tetrahedron .
![$4\sqrt{3}S[ABCD] = 4\sqrt{3}\left(\sum \sqrt{\left(\frac{a_1+a_2+a_4}{2}\right)\left(\frac{-a_1+a_2+a_4}{2}\right)\left(\frac{a_1-a_2+a_4}{2}\right)\left(\frac{a_1+a_2-a_4}{2}\right)}\right)$](http://latex.artofproblemsolving.com/5/e/f/5efce45883c7d542da3d5f85c5f3a87101d7c830.png)
In fact it suffices to disregard 
and consider each individual sum.
After expanding and rewriting, the inequality is equivalent to 
in the expansion of the RHS, we're done after summing over the four triples.
such that 
.
