# 1993 AJHSME Problems/Problem 23

## Problem

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? $\text{(A)}\ P\text{ and }Q \qquad \text{(B)}\ P\text{ and }R \qquad \text{(C)}\ P\text{ and }S \qquad \text{(D)}\ P\text{ and }T \qquad \text{(E)}\ P,S\text{ and }T$

## Solution

First, note that $P$ must beat $Q$, $R$, $T$, and by transitivity, $S$. Thus $P$ is in first, and not in 3rd. Similarly, $S$ is beaten by $P$, $Q$, and by transitivity, $T$, so $S$ is in fourth or fifth, and not in third. All of the others can be in third, as all of the following sequences show. Each follows all of the assumptions of the problem, and they are in order from first to last: PTQRS, PTRQS, PRTQS. Thus the answer is $\boxed{\text{(C)}\ P\ \text{and}\ S}$.

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