Ben Smith, Peyton Park, $4000, $8000
Jay NcNabb, Thoreau Falls, $3000, $5000
Stefanie Hardy, Summerset, $6000, $4000
From the introduction, the three ended up with a total of $17000 after going into "Final Jeopardy!" with a total of $13000 (clue 5).
By clue 1, in the final round, Ben doubled his money and McNabb added $2000, but the Summerset contestant lost 1/3 of his or her winnings.
By clue 4, Stefanie finished third with $4000 in winnings; she is not Ben, so she is McNabb or the Summerset resident in clue 1.
If she were McNabb, Stefanie then would have had $2,000 after "Double Jeopardy!" (1).
In clue 5, then, we would have:
Ben + Summerset + 2000 = 13000
Ben + Summerset = 11000
Also, in clue 1, we would have:
2*Ben + 2/3*Summerset + 4000 = 17000
2*Ben + 2/3*Summerset = 13000
Solving, we double the first equation and subtract the second:
2*Ben + 2 *Summerset = 22000
-2*Ben - 2/3*Summerset = -13000
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4/3*Summerset = 9000
Summerset = 6750
So the person from Summerset would have had $6750 after "Double Jeopardy!" and $4,500 after "Final Jeopardy!", with Ben having $4250 and $8500 after those two rounds and being the winner.
But by clue 2, the second place contestant would have had $5500--no, Stefanie would have had $4,000 and the Summerset contestant $4500.
Therefore, Stefanie is not McNabb; she is the one from Summerset.
Since she ended up with $4000 (4) after losing 1/3 in "Final Jeopardy!" (1), she would have had $6,000 going into the final round.
In clue 5, then, we would have:
Ben + McNabb + 6000 = 13000
Ben + McNabb = 7000
And in clue 1, we would have:
2*Ben + McNabb + 2000 + 4000 (Stefanie) = 17000
2*Ben + McNabb = 11000
Solving, we double the first equation and subtract the second:
2*Ben + 2*McNabb = 14000
-2*Ben - McNabb = -11000
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McNabb = 3000
So McNabb would have had $3000 and $5000 after "Double Jeopardy!" and "Final Jeopardy!" respectively, and Ben would have had $4000 and $8000.
McNabb is therefore not Ben (1), or Stefanie, so he is Jay.
By clue 2, we have Stefanie Hardy and Ben Smith.
Ben is from Peyton Park and Jay from Thoreau Falls (3).
The contestants on the TV favorite finished as follows:
Ben Smith, Peyton Park, $4000, $8000
Jay NcNabb, Thoreau Falls, $3000, $5000
Stefanie Hardy, Summerset, $6000, $4000
Update:
The solution above is the one I originally intended for the problem. During its submission and the editing process, an important piece of information was mistakenly dropped from the problem, allowing for another valid solution (see my further remarks here).
The part that was dropped was that the amounts that the three won totaled to $17000. Were this included, the above answer stands as the unique solution.
With this statement omitted, my first line of reasoning (in which I show that Stefanie cannot be McNabb) becomes invalid, and allows for another possibility. Also, the solution cannot be reached in exactly the same way as I have done it, but of course removing the limitation on the total winnings does not invalidate the original solution.
So, there are two solutions to this problem:
Ben Smith, Peyton Park, $4000, $8000
Jay NcNabb, Thoreau Falls, $3000, $5000
Stefanie Hardy, Summerset, $6000, $4000
and:
Ben Smith from Peyton Park: 3875 to 7750
Jay Hardy of Summerset: 7125 to 4750
Stefanie McNabb of Thoreau Falls: 2000 to 4000
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