[/quote]
,
is lovely if there exist a positive integer 
and (not necessarily distinct) positive integers 
,
,
,
such that 
and 
for 
.
,
be an odd prime. Prove that:![\[\displaystyle\sum_{k=1}^{p-1}k^{2p-1} \equiv \frac{p(p+1)}{2} \pmod{p^2}\]](http://latex.artofproblemsolving.com/a/5/5/a55e75d4d9769ceec3a3ce359aac676c7c7875fd.png)
,
,
,
'![\[n=a_1+a_2+\dots+a_m,\]](http://latex.artofproblemsolving.com/6/c/2/6c2a979a5ff6c40bf2d7152c8c32088fc36f848b.png)
where 
,
,
,
.
![\[
\begin{cases}
x - y z = 1,\\[2pt]
y - z x = 2,\\[2pt]
z - x y = 4.
\end{cases}
\]](http://latex.artofproblemsolving.com/b/2/d/b2d94e49491a71f86109103f985fe3f238cf4d95.png)
be a function such that
for all 
.
.
such that the equality 
holds for all pairs of real numbers 
.
is an obvious solution. Now, let 
.
or 
.
.
with integer coefficients such that 
and 
is a square of an integer for all nonnegative integers 
.
is defined as follows: 
if 
and 
if 
.
,
be all positive integers smaller than 
such that 
for all 
is the largest positive integer that divides both 
and 
.
.
and 
such that 
.
is a benefactor of 
because 
.
benefactors.
denotes the least natural such that
Find all naturals 
such that 
.
Y by john0512
9Poll:
Do you prefer RSM (Russian School of Math) or AoPS?
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I'm doing RSM right now, but I want to switch over to AoPS. I'm just curious to know, which do you prefer? (If you have no experience with both, please don't vote!)
If you want, please drop some advice on the Intermediate Algebra class, because that's the one I'm hoping to take!
If you want, please drop some advice on the Intermediate Algebra class, because that's the one I'm hoping to take!
This post has been edited 1 time. Last edited by BlazingSun1200, Mar 21, 2021, 3:08 AM
