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Sep 15 at 22:49 comment added Timothy Chow @MarkLewko I don't think that tracking down the text of the people who offered the prize is going to lead anywhere. A more promising route may be to look at the collected works of Halphen, to see what problems he stated explicitly.
Sep 15 at 22:44 comment added Mark Lewko @TimothyChow: do you have any information about the original source of this problem? As an aside, I had never heard of the Steiner Prize. Apparently for some of its existence (which came to an end with WWI) eminent mathematicians of the era (including Dirichlet and Weierstrass) would pose problems, with the idea the prize would go to the best work on the problem. Reportedly these problems frequently went unsolved and the prizes unclaimed. It would be interesting to know if the full statements of these problems survive. See: ams.org/bookstore/pspdf/mprize-prev.pdf
Sep 15 at 13:13 comment added Alexandre Eremenko Look at the reference I gave hsm.stackexchange.com/questions/14748/… for examples "outside physics" and older than those in this list.
Sep 14 at 20:06 history answered Timothy Chow CC BY-SA 4.0