[/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
Hi everyone!
I noticed that every time i try to answer a question and get it right/wrong the screen freezes and i can't scroll anymore...
100%
Chrome
PC
do u guys have the same issue?
I don't think it is like this for Alcumus but for normal class problems it does
*the challenge problems
EDIT: sry for the terrible title
I noticed that every time i try to answer a question and get it right/wrong the screen freezes and i can't scroll anymore...
100%
Chrome
PC
do u guys have the same issue?
I don't think it is like this for Alcumus but for normal class problems it does
*the challenge problems
EDIT: sry for the terrible title
This post has been edited 1 time. Last edited by jkim0656, Mar 27, 2025, 2:24 AM
)
