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.
such that![\[f(x+yf(x))+f(xf(y)-y)=f(x)-f(y)+2xy\]](http://latex.artofproblemsolving.com/e/b/c/ebccae7e765d9fc3c8605e58a7399a70127b39b1.png)
and natural number 
,
, such that 
Prove that :
equality cases : ?
distinct positive integers such that adding the product of any two of them to 
yields a perfect square?
be a sequence of positive integers with 
, 
and![\[a_n = a_{n-1}^{a_{n-1}a_{n-2}}-1\]](http://latex.artofproblemsolving.com/1/b/f/1bf1a88979e7d2b33530de979d66a4e81032953d.png)
for all 
. Show that if 
is a prime less than 
for some positive integer 
, then there exists 
such that 
.
for which
be an acute angled scalene triangle with circumcircle 
. Let 
and 
be the circumcenter and orthocenter of 
respectively. Let 
and 
be points on segments 
and , respectively, such that
Prove that 
and 
intersect on the circumcircle of 
.
grid with 
rows and columns, and each square of the grid is filled with a non-negative integer. Let 
be the average of all the numbers in the grid. Prove that if 
the there exist two identical rows (meaning same numbers in the same order).
is called feirense if it is possible to write on a sheet of paper some integers such that every positive divisor of less than is the difference between two numbers on the sheet, and no other positive number is.
and 
are points on the segments 
and 
, respectively, such that 
and 
. If the area of triangle 
is 
, then what is the area of triangle 
?
![$\frac{[AXY]}{[BXY]}=\frac{AX}{BX}=\frac12$](http://latex.artofproblemsolving.com/2/a/9/2a915730c9421688ecd5dd77f0f60dfe0d3cda51.png)
(Collinear Bases with common vertex (since their altitude is the same, the ratio of area is equal to the ratio of bases))![$[BXY]=20,[BYA]=30$](http://latex.artofproblemsolving.com/c/1/1/c11a0d75f8c6afb22d9554224f6486925cd1e0bd.png)
![$\frac{[BYA]}{[BYC]}=\frac{AY}{YC}=\frac21$](http://latex.artofproblemsolving.com/7/9/5/7953177214258046e77563174589ae9bf1d29dcc.png)
![$[BYC]=15$](http://latex.artofproblemsolving.com/c/0/a/c0a16d4aee2e2d6e983ffc7cd18a78f83d9a3f89.png)
![$[ABC]=10+20+15 = 45$](http://latex.artofproblemsolving.com/0/0/8/008536cc0eaa4126052ed67d163c57fc2ff24788.png)
(Collinear Bases with common vertex (since their altitude is the same, the ratio of area is equal to the ratio of bases))
satisfy that: 
![$$[AXY]=\frac12 AY \cdot AX \cdot \sin\angle XAY$$](http://latex.artofproblemsolving.com/c/1/6/c164ab44d74478c0782b3ba13e3ec2b2f33d05da.png)
and ![$$[ABC]=\frac12 AC\cdot AB \cdot \sin\angle BAC=\frac92 AC\cdot AB \cdot \sin\angle XAY.$$](http://latex.artofproblemsolving.com/e/8/9/e89845bd6394a47149e5a69ea81809fead89b7b9.png)
From this, we conclude ![$[ABC]=9\cdot [AXY]=\boxed{90}$](http://latex.artofproblemsolving.com/0/7/a/07ad2b699249fc61c924e6dd687410f28213d9cf.png)
as the answer