MathDB

3

Part of 2016 Romania Team Selection Tests

Problems(3)

Romania TST 2016 Day 1 P3

Source: Romania TST 2016 Day 1 P3

11/1/2017
Let nn be a positive integer, and let a1,a2,..,ana_1,a_2,..,a_n be pairwise distinct positive integers. Show that ∑k=1n1[a1,a2,…,ak]<4,\sum_{k=1}^{n}{\frac{1}{[a_1,a_2,…,a_k]}} <4, where [a1,a2,…,ak][a_1,a_2,…,a_k] is the least common multiple of the integers a1,a2,…,aka_1,a_2,…,a_k.
number theorynumber divisors
Romania TST 2016 Day 2 P3

Source: Romania TST 2016 Day 2 P3

11/1/2017
Prove that: (a) If (an)n≥1(a_n)_{n\geq 1} is a strictly increasing sequence of positive integers such that a2n−1+a2nan\frac{a_{2n-1}+a_{2n}}{a_n} is a constant as nn runs through all positive integers, then this constant is an integer greater than or equal to 44; and (b) Given an integer N≥4N\geq 4, there exists a strictly increasing sequene (an)n≥1(a_n)_{n\geq 1} of positive integers such that a2n−1+a2nan=N\frac{a_{2n-1}+a_{2n}}{a_n}=N for all indices nn.
number theoryalgebra
Romania TST 2016 Day 3 P3

Source: Romania TST 2016 Day 3 P3

11/1/2017
Given a positive integer nn, show that for no set of integers modulo nn, whose size exceeds 1+n+41+\sqrt{n+4}, is it possible that the pairwise sums of unordered pairs be all distinct.
combinatorics