4

Part of 2024 Serbia Team Selection Test

Problems(2)

A bijection with close adjacent values

Source: Serbia TST 2024, P4

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

be a bijection and let

k

be a positive integer such that

|f(x+1)-f(x)| \leq k

for all positive integers

x

. Show that there exists an integer

d

f(x)=x+d

for infinitely many positive integers

x

Factorial equation

Source: Serbia Additional IMO TST 2024, P4 (out of 4)

n!_0

denote the number obtained from

n!

by deleting all the zeroes in the end of it decimal representation. Find all positive integers

a, b, c

a!_0+b!_0=c!_0