be a positive integer. Ana and Banana play a game. Banana thinks of a function 
and a prime number 
.
is nonconstant, 
,
for all integers 
.
.
in order. Ana wins if she can determine the value of 
Let 
be positive reals such that 
.
of positive integers is called central if for every positive integer 
terms of the sequence is equal to 
of positive integers such that for every central sequence 
there are infinitely many positive integers 
.
Prove that
be a set of 
points in the plane, no three of which are collinear. Initially these points are connected with 
and 
sharing a common interior point and replace them by the segments 
and 
if none of them is present at this moment. Prove that it is impossible to perform 
or more such moves.
is the center of the inscribed circle and holds 
.
.
and 
.
and 
.
and 
intersect the circumcircle of 
and 
,
and 
such that 
and 
are parallelograms (with 
,
)
be the point of intersection of lines 
and 
.
,
,
be all positive integers smaller than 
such that 
for all 
is the largest positive integer that divides both 
and 
.
.
,
,
is the point such that 
and 
,
lie on the same side of 
be the point on 
,
.
passes through the midpoint of 
Y by Adventure10, Mango247
Let 
be an equilateral triangle with side lenght is 
.Let ![$D \in [AB]$](//latex.artofproblemsolving.com/0/7/9/07914568284f3822cf15730a6a0e868f0bd79996.png)
is a point. Perpendiculars from 
to ![$[AC]$](//latex.artofproblemsolving.com/0/9/3/0936990e6625d65357ca51006c08c9fe3e04ba0c.png)
and ![$[BC]$](//latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
intersects with and at points 
and 
respectively. Perpendiculars from and to ![$[AB]$](//latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
intersects with at points 
and 
. Prove that
![$$[E_1F_1]=\frac{3}{4}$$](//latex.artofproblemsolving.com/3/e/f/3efdf53cb3ff4b2a7fa60691f5aac9b3eeecf00c.png)
be an equilateral triangle with side lenght is 
.Let ![$D \in [AB]$](http://latex.artofproblemsolving.com/0/7/9/07914568284f3822cf15730a6a0e868f0bd79996.png)
is a point. Perpendiculars from 
to ![$[AC]$](http://latex.artofproblemsolving.com/0/9/3/0936990e6625d65357ca51006c08c9fe3e04ba0c.png)
and ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
intersects with and at points 
and 
respectively. Perpendiculars from and to ![$[AB]$](http://latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
intersects with at points 
and 
. Prove that![$$[E_1F_1]=\frac{3}{4}$$](http://latex.artofproblemsolving.com/3/e/f/3efdf53cb3ff4b2a7fa60691f5aac9b3eeecf00c.png)
This post has been edited 1 time. Last edited by Emirhan, Jan 30, 2016, 6:01 PM
Reason: 404
Reason: 404
.
.
and 
.
.
