MathDB

5

Part of 2023 Iran Team Selection Test

Problems(2)

function from Z to R

Source: Iran TST 2023 ; Exam 1 Problem 5

3/15/2023
Suppose that n≥2n\ge2 and a1,a2,...,ana_1,a_2,...,a_n are natural numbers that (a1,a2,...,an)=1 (a_1,a_2,...,a_n)=1. Find all strictly increasing function f:Z→Rf: \mathbb{Z} \to \mathbb{R} that:
∀x1,x2,...,xn∈Z:f(∑i=1nxiai)=∑i=1nf(xiai) \forall x_1,x_2,...,x_n \in \mathbb{Z} : f(\sum_{i=1}^{n} {x_ia_i}) = \sum_{i=1}^{n} {f(x_ia_i})
Proposed by Navid Safaei and Ali Mirzaei
functionalgebra
two polynomial with real root

Source: Iran TST 2023 ; Exam 2 Problem 5

3/16/2023
Find all injective f:Z≥0→Z≥0f:\mathbb{Z}\ge0 \to \mathbb{Z}\ge0 that for every natural number nn and real numbers a0,a1,...,ana_0,a_1,...,a_n (not everyone equal to 00), polynomial ∑i=0naixi\sum_{i=0}^{n}{a_i x^i} have real root if and only if ∑i=0naixf(i)\sum_{i=0}^{n}{a_i x^{f(i)}} have real root.
Proposed by Hesam Rajabzadeh
algebrapolynomialfunction