CombinatoricsDifficulty 5.2AIME, harderFind the answerItaly
Problem:
After a race among five horses, five friends meet and talk about the results. It is known that each of them bet on a different horse, and that both the person who bet on the horse that finished first and the person who bet on the horse that finished last are lying; the others are telling the truth. Their statements are as follows:
Alex: "The horse Igor bet on finished at least two positions ahead of Enrica's horse."
Enrica: "The horse I bet on won."
Igor: "The horse Osvaldo bet on beat mine."
Osvaldo: "The horse I bet on did not finish among the first three."
Umberto: "My horse did not win but finished immediately after Alex's and immediately before Enrica's."
Who bet on the horse that finished third?
Pick one
Solution
Solution:
The answer is (A). Enrica cannot but be lying (if her horse had finished first as she claims, she would have to be lying, contradiction) and so her horse is last. If Umberto were telling the truth his horse would be second-to-last, and Alex's would be third. In that case Osvaldo is lying (the last two positions are already occupied) and so his horse is the winner; Igor's horse is second, and both Igor and Alex are telling the truth. If instead Umberto were lying he would have to have bet on the winning horse; Osvaldo would have to have bet on the horse that finished fourth in order not to be on the podium, but then Igor would have to be lying, contradiction.
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