MathDB

2

Part of 2020 European Mathematical Cup

Problems(2)

Fibby numbers

Source: European Mathematical Cup 2020, Problem J2

12/22/2020
A positive integer k⩾3k\geqslant 3 is called fibby if there exists a positive integer nn and positive integers d1<d2<…<dkd_1 < d_2 < \ldots < d_k with the following properties: \\ ∙\bullet dj+2=dj+1+djd_{j+2}=d_{j+1}+d_j for every jj satisfying 1⩽j⩽k−21\leqslant j \leqslant k-2, \\ ∙\bullet d1,d2,…,dkd_1, d_2, \ldots, d_k are divisors of nn, \\ ∙\bullet any other divisor of nn is either less than d1d_1 or greater than dkd_k.
Find all fibby numbers. \\ \\ Proposed by Ivan Novak.
number theoryFibonaccidivisorSequenceemc
permutations and k-mutations

Source: 9th EMC, 12th December 2020 - 20th December 2020. SENIOR league, P2.

12/22/2020
Let nn and kk be positive integers. An nn-tuple (a1,a2,…,an)(a_1, a_2,\ldots , a_n) is called a permutation if every number from the set {1,2,...,n}\{1, 2, . . . , n\} occurs in it exactly once. For a permutation (p1,p2,...,pn)(p_1, p_2, . . . , p_n), we define its kk-mutation to be the nn-tuple (p1+p1+k,p2+p2+k,...,pn+pn+k),(p_1 + p_{1+k}, p_2 + p_{2+k}, . . . , p_n + p_{n+k}), where indices are taken modulo nn. Find all pairs (n,k)(n, k) such that every two distinct permutations have distinct kk-mutations.
Remark: For example, when (n,k)=(4,2)(n, k) = (4, 2), the 22-mutation of (1,2,4,3)(1, 2, 4, 3) is (1+4,2+3,4+1,3+2)=(5,5,5,5)(1 + 4, 2 + 3, 4 + 1, 3 + 2) = (5, 5, 5, 5).
Proposed by Borna Šimić