be a positive integer. There are two numbers written on a blackboard, one red and one blue. Initially, both are 0. We define the following procedure: at each step, we choose a nonnegative integer 
(not necessarily distinct from the previously chosen ones), and, if the red and blue numbers are 
and 
respectively, we replace them with 
and 
,
.
are positiv real number, such that 
prove inequality 
and 
.
Y by
Consider a parallelogram ![$[ABCD]$](//latex.artofproblemsolving.com/c/d/3/cd3361fb9586094387d885d23e9efce948e8b291.png)
such that 
is an acute angle. Let 
be a point in line 
different from 
such that 
, and let 
be a point in line 
different from such that 
. Prove that triangle ![$[GDH]$](//latex.artofproblemsolving.com/b/b/6/bb6cfccbfd130c40e05b1c52b8eab8f8b80c0e43.png)
is isosceles.
![$[ABCD]$](http://latex.artofproblemsolving.com/c/d/3/cd3361fb9586094387d885d23e9efce948e8b291.png)
such that 
is an acute angle. Let 
be a point in line 
different from 
such that 
, and let 
be a point in line 
different from such that 
. Prove that triangle ![$[GDH]$](http://latex.artofproblemsolving.com/b/b/6/bb6cfccbfd130c40e05b1c52b8eab8f8b80c0e43.png)
is isosceles.This post has been edited 1 time. Last edited by parmenides51, Oct 9, 2020, 5:27 PM
