[/quote]
,
distinct divisors of a positive integer number 
form an arithmetic progression. Prove that the number 
divisors.
satisfy that: 
.
be distinct real numbers such that 
Find the value of 
Find the value of 
Find the value of 
be a positive integer. A positive integer 
digits in decimal representation (it cannot have leading zeros).
is a ![\[(20 + 25)^2 = 2025.\]](http://latex.artofproblemsolving.com/5/c/5/5c51bca485cea6c3750c48c09091eb5fd72b5192.png)
Find all 
be the set of real numbers. Let 
be a function such that ![\[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\]](http://latex.artofproblemsolving.com/5/d/3/5d3476fd9195d085d0030d6036a9fde1cf6756a7.png)
for every 
.
.
for every 
or 
for every 
such that, for all integers 
,![\[f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).\]](http://latex.artofproblemsolving.com/a/9/e/a9edf1a3207d29facc970b6ee288d4fabdd43a4b.png)
denotes the set of integers.)
with degree 
is reflexive if there is an integer 
such that 
for every 
,
for 
.
be an integer and 
be a polynomial with integer coefficients. Prove that there exist reflexive polynomials 
,
with integer coefficients such that![\[(1+x+x^2+\dots+x^{\ell-1})p(x)=q(x)+x^\ell r(x)\]](http://latex.artofproblemsolving.com/6/6/7/66729b54a8b87c6ddbfdf7e7e5932dbb8b262439.png)
and 
.
and 
.
be a natural number. We write out the fractions 
,
,
, 
such that they are all in their simplest form. Let the sum of the numerators be 
.
is one of 
odd, but the other is even?
Y by jhu08, Adventure10, Mango247
1- Find a pair 
of positive integers such that 
in all of this cases :
2-Is there a pair of positive integers such that : 
3-Every prime number less than 
can be represented in the form 
by taking an Appropriate pair
of positive integers. Prove that the number cannot be represented in the form where 
and 
are positive integers
4-Note that
. The number 
is the least prime number that cannot be represented as a sum or an difference of a power of 
and a power of 
. Prove that the number cannot be represented in any of the forms 
, 
, where and are positive integers
of positive integers such that 
in all of this cases :
2-Is there a pair
of positive integers such that : 
3-Every prime number less than 
can be represented in the form 
by taking an Appropriate pair of positive integers. Prove that the number
cannot be represented in the form where 
and 
are positive integers4-Note that
. The number 
is the least prime number that cannot be represented as a sum or an difference of a power of 
and a power of 
. Prove that the number cannot be represented in any of the forms 
, 
, where and are positive integers
then 
which is not possible for any integer 
, and can be easily checked by modular arithmetic. since, 
and 
. So, just check 
but 
or 
in both cases it does not work.
if 
in mod 
but 
for some 
:
or 
in both cases it does not work.
