AoPS Academy
be positive reals. Prove that: 
rabbits and carrots. Each rabbit will choose randomly a carrot to eat. Find the probability of a carrot was chose by less than 
rabbit.
and 
be positive rational numbers such that: 
(
, 
, 
) with excircle 
and incircle 
. Incircle touches 
at 
. The excircle with the diameter of 
cuts excircle at 
(
). 
cuts the excircle with the diameter of at 
(
) and 
cuts 
at 
. Prove that:
~ 
and is the bisector of 
is the tangent line of the excircle of 
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))
![$$[AXY]=\frac12 AY \cdot AX \cdot \sin\angle XAY$$](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