4

Part of 2021 Iran Team Selection Test

Problems(2)

Easy problem to give us comfort

Source: Iranian TST 2021, second exam day 2, problem 1

Find all functions

f : \mathbb{N} \rightarrow \mathbb{R}

such that for all triples

a,b,c

of positive integers the following holds :

f(ac)+f(bc)-f(c)f(ab) \ge 1

Proposed by Mojtaba Zare

algebrainequalitiesfunctionnumber theory

Random prime functions on polynomials

Source: Iranian TST 2021, first exam day 2, problem 4

\Omega(n),\omega(n)

be the biggest and smallest prime factors of

n

respectively . Alireza and Amin decided to play a game. First Alireza chooses

1400

polynomials with integer coefficients. Now Amin chooses

700

of them, the set of polynomials of Alireza and Amin are

B,A

respectively . Amin wins if for all

n

\max_{P \in A}(\Omega(P(n))) \ge \min_{P \in B}(\omega(P(n)))

Who has the winning strategy.
Proposed by Alireza Haghi

number theorypolynomialprime numbersfunctionalgebraCombi