[/quote]
,
natural numbers around the perimeter of a circle. She wants the 
Can she accomplish this?
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 
,
.
,
with incenter 
,
,
-
.
be the orthocenter of 
and let 
.
be the point on 
and the line 
meets 
.
and parallel to 
bisects 
.
be a point inside the triangle 
intersects 
at 
.
and 
are defined similarly. Line 
intersects 
at points 
,
.
,
.
,
,
,
are defined similarly. Prove that points 
and 
. Prove that
. Prove that 
and 
. Prove that
Where 
![$ a,b\in [0,1] $](http://latex.artofproblemsolving.com/9/7/5/975edfc0cf5316d78283b313c6b5cfbdd7b92b6e.png)
. Prove that
Where 
Where 
where 
are lengths of parallel sides and 
are other two sides.
be real numbers satisfying 
and ![\[a_{i+1} - 2a_i + a_{i-1} = a^2_i\]](http://latex.artofproblemsolving.com/c/d/8/cd878e4ac230853182da75c54dcbf7be05b305bc.png)
for 
Prove that 
for 
Y by Adventure10, Mango247, and 1 other user
The side lengths 
of a triangle 
form an arithmetical progression (such that 
). The side lengths 
of a triangle 
also form an arithmetical progression (with 
). [Hereby, 
] Moreover, we know that 
Show that triangles and are similar.
of a triangle 
form an arithmetical progression (such that 
). The side lengths 
of a triangle 
also form an arithmetical progression (with 
). [Hereby, 
] Moreover, we know that 
Show that triangles
and are similar.This post has been edited 1 time. Last edited by ahaanomegas, Dec 26, 2014, 9:29 PM
Reason: I have changed the wording of the problem since the conditions given in the original were not sufficient, at least not in the form they were written. - darij
Reason: I have changed the wording of the problem since the conditions given in the original were not sufficient, at least not in the form they were written. - darij
s.t. it still keeps the same shape, but 
.
.
are collinear. We can assume WLOG that 
.
,
,
to match 
also stretches 
to match 
,
are similar.
