AoPS Academy
be a triangle, 
its orthocenter, 
its circumcenter, and 
its circumradius. Let 
be the reflection of the point 
across the line 
, let 
be the reflection of the point 
across the line 
, and let 
be the reflection of the point 
across the line 
. Prove that the points , and are collinear if and only if 
.
, 
, 
, 
, 
, 
initially contains one coin. The following operations are allowed
, 
, remove one coin from and add two coins to 
;
, 
, remove one coin from and swap the contents (maybe empty) of the boxes 
and 
., , , , become empty, while box contains exactly 
coins.
be a real number and let 
be arbitrary real numbers in 
. The sequences 
and 
are then defined recursively by
. Prove that 
for all sufficiently large 
.
is a prime number
. So we try making a linear function of the form 
. But notice![$$r_a + r_b + r_c = 4R \sin \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2} + 4R \cos \frac{A}{2} \sin \frac{B}{2} \cos \frac{C}{2} + 4R \cos \frac{A}{2} \cos \frac{B}{2} \sin \frac{C}{2} = 4R [\sin (\frac{A}{2} + \frac{B}{2} + \frac{C}{2}) + \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}] = 4R + r$$](http://latex.artofproblemsolving.com/4/3/1/431b1221f7b43823df6589a25c0c2e1ec0757801.png)
of five variables such that, for any triangle 
,
we have 
?
