Problem 6. Let $p>2$ be prime. Find the number of the subsets $B$ of the set $A=\{1,2,\ldots,p-1\}$ such that, the sum of the elements of $B$ is divisible by $p.$ Ivan Landgev
Problem
Source: Bulgarian TST1/2006 Problem 6
Tags: algebra, polynomial, combinatorics unsolved, combinatorics
bilarev
31.05.2006 20:14
See http://www.mathlinks.ro/Forum/viewtopic.php?t=15112 . It is the same because you can find the number of $l-element$ subsets of $A$ with the condition for every $l$(consider the polynomial $P(x)=\left(x-\epsilon\right)\left(x-\epsilon^2\right)...\left(x-\epsilon^{p-1}\right)$, $\epsilon$ is $p^{th}$ root of unity!).
Xixas
31.05.2006 22:11
Indeed, the problem is well known! It's strange that it appeared in an IMO TST...
imortal
31.05.2006 22:36
Yes, I think that it was on Sankt-Peterburg Olympiad in 2004 also.
José
01.06.2006 03:28
so strange!
egxa
15.07.2023 15:05
bilarev wrote: http://www.mathlinks.ro/Forum/viewtopic.php?t=15112 This link isn't working. Can anyone post the solution of the problem.
megarnie
15.07.2023 16:07
The link is now https://artofproblemsolving.com/community/c6h15112
egxa
15.07.2023 22:23
megarnie wrote: The link is now https://artofproblemsolving.com/community/c6h15112 thx
somebodyyouusedtoknow
15.07.2023 22:30
Also British MO 2019 P3, and Indonesian First Stage TST for IMO 2021