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.
where 
are lengths of parallel sides and 
are other two sides.
be real numbers satisfying 
and ![\[a_{i+1} - 2a_i + a_{i-1} = a^2_i\]](http://latex.artofproblemsolving.com/c/d/8/cd878e4ac230853182da75c54dcbf7be05b305bc.png)
for 
Prove that 
for 
, such that one of the 
letters 
can be placed in each cell of a table with rows and 
columns, and has the following property: For any two distinct rows in the table, there exists at most one column, such that the entries of these two rows in such a column are the same letter.
be an continuous function defined in ![$\text{[0,2015]},f(0)=f(2015)$](http://latex.artofproblemsolving.com/c/9/9/c99f63e726b504aea8c1a65ba2926155a873effa.png)
pairs of 
such that 
such that ![\[f(f(x)) + xf(xy) = x + f(y)\]](http://latex.artofproblemsolving.com/a/6/f/a6f84dd53eec985b90bad3df6580eeaf29d73195.png)
for all positive real numbers 
and 
.
be a triangle with 
. Let the incenter and incircle of triangle be 
and 
, respectively. Let 
be the point on line 
different from 
such that the line through parallel to 
is tangent to . Similarly, let 
be the point on line different from 
such that the line through parallel to 
is tangent to . Let 
intersect the circumcircle of triangle at 
. Let 
and 
be the midpoints of and , respectively.
.
the bisector of angle 
intersects at 
. Show that if 
is equal to the radius of the circle circumscribed by , then the following are verified:
is strictly greater than 
and strictly less than 
.
the bisector of angle intersects at . Show that if is equal to the radius of the circle circumscribed by , then the following are verified: is strictly greater than and strictly less than .
and 
. Prove that
. Prove that 
and 
. Prove that
Where 
![$ a,b\in [0,1] $](http://latex.artofproblemsolving.com/9/7/5/975edfc0cf5316d78283b313c6b5cfbdd7b92b6e.png)
. Prove that
Where 
Where 
