[/quote]
,
be a triangle with circumcircle 
and incenter 
and let 
be the midpoint of 
.
,
such that 
,
,
.
intersects 
other than 
and 
meet on 
and 
Prove that
,
,
,
,
,
initially contains one coin. The following operations are allowed
,
,
;
,
,
and 
.
coins.
Prove that
its orthocenter, 
its circumcenter, and 
its circumradius. Let 
,
.
be a real number and let 
be arbitrary real numbers in 
.
and 
are then defined recursively by
.
for all sufficiently large 
.
is a prime number
is inscribed in circle 
.
its 
denote the midpoint of arc 
.
meets 
.
at 
and 
meets 
a second time at 
,
meets 
a second time at 
.
passes through the antipodal to 
intersects the lines 
,
at points 
,
,
of the segments 
,
,
,
.
be a positive integer. Prove that there are infinitely many primes, each of which divides at least one term of the sequence 
,
.
Y by
Miguel has a square piece of paper 
that he folded along a line 
, 
on 
, and 
on 
. This fold sent 
to point 
on 
, distinct from 
and 
. Also, it brought 
to point 
. 
is the intersection of 
and 
. Prove that the inradius of 
is equal to the sum of the inradius of 
and 
.
that he folded along a line 
, 
on 
, and 
on 
. This fold sent 
to point 
on 
, distinct from 
and 
. Also, it brought 
to point 
. 
is the intersection of 
and 
. Prove that the inradius of 
is equal to the sum of the inradius of 
and 
.
.
,
.
.
.
.
.![$D'F\ :\ y-2=\frac{2\lambda}{1-\lambda^{2}}[x-(\lambda-1)^{2}]$](http://latex.artofproblemsolving.com/f/3/3/f3391df237bc362cfba9377f655a952c40461e3c.png)
.
.![$A'E\ :\ y=\frac{2\lambda}{1-\lambda^{2}}[x-(\lambda^{2}+1)]$](http://latex.artofproblemsolving.com/8/f/5/8f57e4779aa8e13f22ac4a59a572c3e55635d9c7.png)
.
.
.
