MathDB

1

Part of 2018 Danube Mathematical Competition

Problems(2)

d_i are all the proper divisors of n, d_i+1 are all the proper divisors of m

Source: Danube 2018 junior p1

7/22/2019
Find all the pairs (n,m)(n, m) of positive integers which fulfil simultaneously the conditions: i) the number nn is composite; ii) if the numbers d1,d2,...,dk,k∈N∗d_1, d_2, ..., d_k, k \in N^* are all the proper divisors of nn, then the numbers d1+1,d2+1,...,dk+1d_1 + 1, d_2 + 1, . . . , d_k + 1 are all the proper divisors of mm.
number theoryDivisorsComposite
\sum_{i=1}^{k} x_i \le k - 1, inequality with labeling beads from necklace

Source: Danube 2018 p1

7/22/2019
Suppose we have a necklace of nn beads. Each bead is labeled with an integer and the sum of all these labels is n−1n - 1. Prove that we can cut the necklace to form a string, whose consecutive labels x1,x2,...,xnx_1,x_2,...,x_n satisfy ∑i=1kxi≤k−1\sum_{i=1}^{k} x_i \le k - 1 for any k=1,...,nk = 1,...,n
inequalitiescombinatorics