MathDB

5

Part of 2017 QEDMO 15th

Problems(2)

s^n (F) = F U {a + n|a \in F }

Source: 15th -a QEDMO problem 5 (19. - 22. 10. 2017) https://artofproblemsolving.com/community/c1512515_qedmo_2005

5/29/2021
Let FF be a finite subset of the integer numbers. We define a new subset s(F)s(F) in that a∈Za\in Z lies in s(F)s (F) if and only if exactly one of the numbers aa and a−1a -1 in FF. In the same way one gets from s(F)s (F) the set s2(F)=s(s(F))s^2(F) = s (s (F)) and by nn-fold application of ss then iteratively further subsets sn(F)s^n (F). Prove there are infinitely many natural numbers nn for which sn(F)=F∪{a+n∣a∈F}s^n (F) = F\cup \{a + n|a \in F\}.
number theory
f (x) = x^n + x^{n-1} +...+ x + 1= g (h (x))

Source: 15th -b QEDMO problem 5 (19. - 22. 10. 2017) https://artofproblemsolving.com/community/c1512515_qedmo_2005

5/30/2021
For which natural numbers nn can the polynomial f(x)=xn+xn−1+...+x+1f (x) = x^n + x^{n-1} +...+ x + 1 as write f(x)=g(h(x))f (x) = g (h (x)), where gg and hh should be real polynomials of degrees greater than 11?
algebrapolynomial