I suspect that there have been many instances throughout history where a new proof of an existing result has been discovered by a student while taking an exam. Does anyone have an example of this?
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3$\begingroup$ For questions that, by design, do not have a single right answer, I think it is usual to flag for conversion to community wiki. $\endgroup$– LSpiceCommented Dec 26, 2022 at 22:17
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1$\begingroup$ I would suggest to specify the requirement that answers contain an explicit reference, to avoid the propagation of urban legends. $\endgroup$– Federico PoloniCommented Dec 26, 2022 at 23:32
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1$\begingroup$ A possible answer is Problem 6 of the 1988 Mathematics Olympiad. There is circumstantial evidence that Emanouil Atanassov's solution came as a surprise. (I asked on the HSM StackExchange about this origin of the IMO problem but did not receive any answers.) Atanassov's method is now typically called Vieta jumping. As explained on math.SE, the method is implicit in the theory of symmetries of conics, but arguably not explicit. $\endgroup$– Timothy ChowCommented Dec 27, 2022 at 5:14
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3$\begingroup$ More generally, if you want to find examples, the list of IMO special prizes might be a good starting point. These are awarded for particularly beautiful solutions, and some of them might count as a "new proof of an existing result." But I don't know if the prize-winning solutions are published anywhere. $\endgroup$– Timothy ChowCommented Dec 27, 2022 at 5:22
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In 1986, George Bernard Dantzig himself, told us a fact about his younger days at the Berkeley University, when he was just a PhD candidate (1939).
One day, he arrived too late to attend his advanced statistics class, so he took a note of the pair of "exercises" that he found on the blackboard, thinking that, as usual, they would have been done by candidates as a standard homework.
At home, after some hard work, he finally managed to solve them, providing a full proof. Then he gave his manuscript to his professor. He took it, without providing useful comments, but later, about six weeks later, Dantzig received a surprising feedback from that professor, since he told him that, in the meantime, he had written an article based on Dantzig's solutions to the open problems previously written on the blackboard, announcing that they would have been published very soon.
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3$\begingroup$ This does not seem to be exactly an answer to the original question, which is about new proofs of existing results (not open problems) that were found by students taking exams (not solving, actually or purportedly, homework). I am not sure why these constraints were chosen, or whether either or both is important to the OP, so I guess it will be up to them to say whether this answers their question. $\endgroup$– LSpiceCommented Dec 26, 2022 at 22:17
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$\begingroup$ The second one of those the two problems was independently solved by another author, years later, but Dantzig was added as a coauthor of that paper, since they both used the same approach to its solution. A similar story could be told about Banach's PhD dissertation... "But that is another story and shall be told another time." (cit. Michael Ende). Basically, IMHO, it's hard to be certain about many math myths and stories, so I have decided to mention only a well-known fact, based on a published and serious source. $\endgroup$ Commented Dec 26, 2022 at 22:46