MathDB

2

Part of 2016 Romania Team Selection Tests

Problems(3)

Romania TST 2016 Day 1 P2

Source: Romania TST 2016 Day 1 P2

11/1/2017
Let nn be a positive integer, and let S1,S2,…,SnS_1,S_2,…,S_n be a collection of finite non-empty sets such that ∑1≤i<j≤n∣Si∩Sj∣∣Si∣∣Sj∣<1.\sum_{1\leq i<j\leq n}{\frac{|S_i \cap S_j|}{|S_i||S_j|}} <1. Prove that there exist pairwise distinct elements x1,x2,…,xnx_1,x_2,…,x_n such that xix_i is a member of SiS_i for each index ii.
combinatorics
Romania TST 2016 Day 3 P2

Source: Romania TST 2016 Day 3 P2

11/1/2017
Given a positive integer kk and an integer a≡3(mod8)a\equiv 3 \pmod{8}, show that am+a+2a^m+a+2 is divisible by 2k2^k for some positive integer mm.
number theory
Function in Z^+

Source: Romania TST 2016 Day 5 Problem 2

6/2/2016
Determine all f:Z+→Z+f:\mathbb{Z}^+ \rightarrow \mathbb{Z}^+ such that f(m)≥mf(m)\geq m and f(m+n)∣f(m)+f(n)f(m+n) \mid f(m)+f(n) for all m,n∈Z+m,n\in \mathbb{Z}^+
functionalgebranumber theoryfunctional equation