MathDB

Prove that $2\log_3 n \leq f(n) < 5\log_3 n$ for every $n>1$.

Source: Moldova TST 2001

August 6, 2023

Problem Statement

For every nonnegative integer

n{}

let

f(n)

be the smallest number of digits

1

which can represent the number

n{}

using the symbols

"+", "-", "\times", "(", ")"

. For example,

80=(1+1+1+1+1)\times(1+1+1+1)\times(1+1+1+1)

and

f(80)\leq 13

. Prove that

2\log_3 n \leq f(n) < 5\log_3 n

for every

n>1