Problem

Source: 2017 Iran TST second exam day2 p5

Tags: combinatorics, Iran, Iranian TST



$k,n$ are two arbitrary positive integers. Prove that there exists at least $(k-1)(n-k+1)$ positive integers that can be produced by $n$ number of $k$'s and using only $+,-,\times, \div$ operations and adding parentheses between them, but cannot be produced using $n-1$ number of $k$'s. Proposed by Aryan Tajmir