Question 2
An integer is assigned to each vertex of a regular pentagon
so that the sum of the five integers is $2011$.
A turn of a solitaire game consists of subtracting an integer $m$
(not necessarily positive) from each of the integers at two neighboring vertices
and adding $2m$ to the opposite vertex, which is not adjacent
to either of the first two vertices.
(The amount $m$ and the vertices chosen can vary from turn to turn.)
The game is won at a certain vertex if, after some number of turns,
that vertex has the number $2011$ and the other four vertices have the number $0$.
Prove that for any choice of the initial integers,
there is exactly one vertex at which the game can be won.
so that the sum of the five integers is $2011$.
A turn of a solitaire game consists of subtracting an integer $m$
(not necessarily positive) from each of the integers at two neighboring vertices
and adding $2m$ to the opposite vertex, which is not adjacent
to either of the first two vertices.
(The amount $m$ and the vertices chosen can vary from turn to turn.)
The game is won at a certain vertex if, after some number of turns,
that vertex has the number $2011$ and the other four vertices have the number $0$.
Prove that for any choice of the initial integers,
there is exactly one vertex at which the game can be won.