AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
distinct divisors of a positive integer number 
form an arithmetic progression. Prove that the number has at least 
divisors.
be distinct real numbers such that 
Find the value of 
be distinct real numbers such that 
Find the value of 
be distinct real numbers such that 
Find the value of 
be a positive integer. A positive integer 
is called a -good number if it satisfies has exactly 
digits in decimal representation (it cannot have leading zeros). digits and the last digits of are considered as two separate -digit.
is a 
-good number because![\[(20 + 25)^2 = 2025.\]](http://latex.artofproblemsolving.com/5/c/5/5c51bca485cea6c3750c48c09091eb5fd72b5192.png)
Find all 
-good numbers.
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 
. Assume that the inequality is strict for some 
.
for every 
or 
for every .
such that, for all integers that satisfy 
, the following equality holds:![\[f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).\]](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 
, where 
for 
. Let 
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)
, for which 
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 
. For what 
is one of and 
odd, but the other is even?
.
; hence,
. In this case, our upper bound is
. There are probably several ways to do this, but I'm going to use the fact that
.; hence,. In this case, our upper bound is. There are probably several ways to do this, but I'm going to use the fact that
satisfy that: 
.
and 
.
and 
for each nonnegative integer 
.
is the greatest integer less than or equal to 
,
is a trivial one (just note that 
and you are done.)