3

Part of 2021 Iran Team Selection Test

Problems(2)

Who can guess the polynomials ?

Source: Iranian TST 2021, second exam day 1, problem 3

Prove there exist two relatively prime polynomials

P(x),Q(x)

having integer coefficients and a real number

u>0

such that if for positive integers

a,b,c,d

|\frac{a}{c}-1|^{2021} \le \frac{u}{|d||c|^{1010}}

| (\frac{a}{c})^{2020}-\frac{b}{d}| \le \frac{u}{|d||c|^{1010}}

bP(\frac{a}{c})=dQ(\frac{a}{c})

(Two polynomials are relatively prime if they don't have a common root)
Proposed by Navid Safaii and Alireza Haghi

algebrapolynomialinequalitiesnumber theoryrelatively prime

multiplicative functions on a Tst

Source: Iranian TST 2021, first exam day 1, problem 3

4

positive integers

a,b,c,d

abcd \neq 1

and each pair of them have a GCD of

1

. Two functions

f,g : \mathbb{N} \rightarrow \{0,1\}

are multiplicative functions such that for each positive integer

n

f(an+b)=g(cn+d)

Prove that at least one of the followings hold.

i)

for each positive integer

n

f(an+b)=g(cn+d)=0

ii)

There exists a positive integer

k

such that for all

n

(n,k)=1

g(n)=f(n)=1

f

is multiplicative if for any natural numbers

a,b

f(ab)=f(a)f(b)

)
Proposed by Navid Safaii

functionnumber theory