$\textbf{Who Finishes the Second}$
Adam, Bob, and Charlie are the only three athletes who are competing in a series of track and field events. The first, second and third places in each event are awarded $X$, $Y$ and $Z$ points respectively, where $X > Y > Z$ and all are integers. It is known that
- Adam finishes first with $22$ points overall
- Bob wins the javelin event and finishes with $9$ points overall.
- Charlie also finishes $9$ points overall.
Who finishes second in the $100$-meter dash and why?
$\textbf{Answer}$
Charlie.
$\textbf{Analysis}$
This is a challenging problem. Let's first list what we know. Given $X > Y > Z$, the sum $S=X+Y+Z$ must be at least $6 = 3 + 2 + 1$. Also, because of the total points awarded is $22+9+9=40$, we can conclude that $S$ must be a divisor of $40$. Meanwhile, two events are mentioned (javelin and $100$-meter dash), therefore $S$ must be no more than $40\div 2=20$.
Between $6$ and $20$, there are only three possible values of $S$ which divide $40$: $8$, $10$ and $20$. We will first investigate the case $S=8$. In this case, $5$ events are held in total and there are only two possible combinations of $X$, $Y$, $Z$:
- $1 + 2 + 5$
- $1 + 3 + 4$
In the first case ($1 + 2 + 5$), given Bob wins one event and finishes with $9$ points, he must be placed the third in all the other four events. Meanwhile, Adam must win four first places and one second place in order to finish with $22$ points. Accordingly, Charlie must win four second places and one third place. This arrangement will satisfy the points awarded.
Now, because such an arrangement also implies Adam wins the $100$-meter dash (because he wins all but the javelin), his second place must be javelin. It follows that Charlie must win the second place in the $100$-meter dash. Bingo.
$\textbf{Note}$
Strictly speaking, we will have to investigate all other possibilities to either verify they are impossible, or there exist alternative answers. Here, let's skip these for simplicity.