5

Part of 2020 Iran Team Selection Test

Problems(2)

s(2n) and s(n^2)

Source: Iran TST1 Day 2 P5

k \in \mathbb{Z}

prove that there exist infinite pairs of distinct natural numbers such that \begin{align*} n+s(2n)=m+s(2m) \\ kn+s(n^2)=km+s(m^2). \end{align*} (

s(n)

denotes the sum of digits of

n

.)
Proposed by Mohammadamin Sharifi

number theorysum of digitsIranian TST

Fractional part

Source: Iranian TST 2020, second exam day 2, problem 5

For every positive integer

k>1

prove that there exist a real number

x

so that for every positive integer

n<1398

\left\{x^n\right\}<\left\{x^{n-1}\right\} \Longleftrightarrow k\mid n.

Proposed by Mohammad Amin Sharifi

fractional partalgebraDivisibility