Problem

Source: 2017 Argentina L2 P6

Tags: combinatorics, algebra



In the governor elections, there were three candidates: $A$, $B$, and $C$. In the first round, $A$ received $44\%$ of the votes that were cast between $B$ and $C$. No candidate obtained the majority needed to win in the first round, and $C$ was the one who received the least votes of the three, so there was a runoff between $A$ and $B$. The voters for the runoff were the same as in the first round, except for $p\%$ of those who voted for $C$, who chose not to participate in the runoff; $p$ is an integer, $1 \leqslant p \leqslant 100$. It is also known that all those who voted for $B$ in the first round also voted for him again in the runoff, but it is unknown what those who voted for $A$ in the first round did. A journalist claims that, knowing all this, one can infer with certainty who will win the runoff. Determine for which values of $p$ the journalist is telling the truth. Note: The winner of the runoff is the one who receives more than half of the total votes cast in the runoff.