3

Part of 2023 Iran Team Selection Test

Problems(2)

Arman's good numbers

Source: Iran TST 2023 ; Exam 1 Problem 3

Arman, starting from a number, calculates the sum of the cubes of the digits of that number, and again calculates the sum of the cubes of the digits of the resulting number and continues the same process. Arman calls a number

Good

1

after performing a number of steps. Prove that there is an arithmetic progression of length

1402

of good numbers.
Proposed by Navid Safaei

Three sides of triangle in functional equation

Source: Iran TST 2023 ; Exam 2 Problem 3

Find all function

f: \mathbb{R}^{+} \to \mathbb{R}^{+}

such that for every three real positive number

x,y,z

x+f(y) , f(f(y)) + z , f(f(z))+f(x)

are length of three sides of a triangle and for every postive number

p

, there is a triangle with these sides and perimeter

p

.
Proposed by Amirhossein Zolfaghari

function equationalgebrafunctional equation