Question

Five runners, $P, Q, R, S, T,$ have a race, and $P$ beats $Q,$ $P$ beats $R,$ $Q$ beats $S,$ and $T$ finishes after $P$ and before $Q.$ Who could not have finished third in the race?

$P$ and $Q$

$P$ and $R$

$P$ and $S$

$P$ and $T$

$P, S$ and $T$

Solution:

Since $P$ beats $Q,$ $R,$ and $T,$ and no one beats $P,$ runner $P$ finishes first and so cannot be third.

The clues give the chain $P$ before $T$ before $Q$ before $S.$ So $P, T,$ and $Q$ all finish ahead of $S,$ meaning $S$ is no better than fourth and cannot be third either.

Each of $Q, R, T$ can finish third: for example $P, T, Q, R, S$ puts $Q$ third; $P, R, T, Q, S$ puts $T$ third; and $P, T, R, Q, S$ puts $R$ third. So only $P$ and $S$ cannot be third.

Thus, the correct answer is C .

1993 AMC 8 Problem 23

Solution:

Problem 23 in Other Years