[/quote]
,
with the property that for every real numbers 
and 
it holds that
along the perimeter of a circle, then every word of the length 
consisting of letters 
and 
(i.e. 
,
,
,
,
,
,
,
)
-
(i.e. an ordered 
and 
.
be 
positive integers such that the 
products![\[a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n\]](http://latex.artofproblemsolving.com/0/2/c/02cbff15c4d0118156a9d24be09360facb264d5a.png)
form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.
Prove that
Where 
Prove that
be complex numbers such that 
and 
.
be positive real numbers such that 
and 
,
is defined below. 
Show that for all nonnegative integer 
,
is a positive integer.
![\[\frac {x^{7} - 1}{x - 1} = y^{5} - 1.\]](http://latex.artofproblemsolving.com/7/6/2/762492979b016e0f573b993b67a2b26dfb8888f4.png)
be a prime such that: 
.
Prove that
Where 
Prove that
Let 
Prove that
Y by
(a) Let 
be a monic polynomial so that there exists another real coefficients 
that satisfy
![\[P(x^2-2)=P(x)Q(x)\]](//latex.artofproblemsolving.com/6/9/7/697faa929e4fda7a6e9b1cd97849bd42ffc14306.png)
Determine all complex roots that are possible from
(b) For arbitrary polynomial that satisfies (a), determine whether should have real coefficients or not.
be a monic polynomial so that there exists another real coefficients 
that satisfy![\[P(x^2-2)=P(x)Q(x)\]](http://latex.artofproblemsolving.com/6/9/7/697faa929e4fda7a6e9b1cd97849bd42ffc14306.png)
Determine all complex roots that are possible from (b) For arbitrary polynomial
that satisfies (a), determine whether should have real coefficients or not.
