Mathematics contests before 1800 Aside from well known examples of mathematics contests in 1535 and 1548, what are some other examples before 1800?
Background:  In The History of Mathematics: an Introduction, 3rd edition (1995), David Burton mentions the contests between Nicolo Tartaglia and Antonio Maria Fiore in 1535 (p. 292) and between Tartaglia and Ludovico Ferrari at Milan in 1548 (p. 302).  Burton writes (p. 290):  “It was the practice in those days to treat mathematical  discoveries as personal properties, disclosing neither method nor proof, to prevent their application by others to similar problems. This was because scholarly reputation was largely based on public contests. Not only could an immediate monetary prize be gained by proposing problems beyond the reach of one’s rival, but the outcomes of these challenges strongly influenced academic appointments; at that time, university positions were temporary and subject to renewal based on demonstrated achievement.”
Does there exist a fuller description of these "public contests"?  Is there a published list of examples?  I would especially like to see statements of the problems, along with names, dates, etc.
 A: Here is an overview of Euler's entries to the the annual prize competition of the Paris Academy. He came first 12 out of 15 years in the period 1727-1772.
Ronald Calinger describes Euler's first attempt to win this competition:

Euler soon decided to become a mathematician and theoretical physicist
  rather than a rural Evangelical Reformed pastor. With Bernoulli’s
  backing, he obtained his father assent. In 1726 Leonhard was
  completing graduate studies there, writing papers on the masting of
  ships and on algebraic reciprocal trajectories, apparently his first
  two articles. Although he lacked maritime experience, Leonhard won the
  accessit or second place in the Paris Academy prize competition the next year for his paper on ship masting, losing only to Pierre
  Bouguer, who was becoming France’s leading nautical authority. The
  accessit paper reflects a powerful intuition into physics.

A: Lagrange launched a challenge via the French Academy in the 1780s calling on authors to provide a clarification of the use of infinite and infinitesimal numbers in analysis. Carnot wrote an essay that did not win the competition but became a popular text on the foundations of the calculus.
An earlier challenge was issued anonymously by Pascal and involved the solution of certain problems related to the cycloid curve. A small number of mathematicians submitted solutions including the jesuit Lalouvere who was being informally advised by Fermat (who did not submit a solution himself). The result was of course a victory by Pascal and an embarrassment to Lalouvere who committed some errors along the way.
A: The answers so far have focused on the contest problems set by the Academies of Science in the 18th century. Those were common, and often chosen because someone at the Academy was working on the topic. There's a book by Jeremy L. Caradonna, The Englightenment in Practice, about the Paris Academy's contests. See https://books.google.com/books?isbn=0801464374
But that is different from the public contests Burton describes. It's easy to find such claims in the history of mathematics books (including, alas, mine!), but I wonder to what extent they are just extrapolated from what we know of the Cardano-Tartaglia affair. 
We certainly know that Tartaglia engaged in such contests, because the "Cartelli di Sfida Matematica" were published. They contain the back-and-forth challenges between Tartaglia and various people with links to Cardano. This was issued in facsimile not too many years ago, so it's possible to see exactly what the questions and answers involved. This suggests to me that these contests were more likely conducted in writing, and not as some sort of public debate. But maybe it was both: questions submitted and answered in writing, followed by a public disputation. I'd like to see more evidence. 
It seems unclear, however, to what extent this was "standard practice". Cardano, for example, does not seem to have felt any need to take part in person. Public disputations seem to have happened in this period (the majority of them, perhaps, connected to the Reformation). It seems worthwhile to take a deeper look.
A: Also different from the "public contests" described by Burton, but notable among   mathematical contests before 1800, were those at Cambridge University.  These are described in June Barrow-Green, 'A Corrective to the Spirit of too Exclusively Pure Mathematics': Robert Smith (1689-1768) and his Prizes at Cambridge University, Annals of Science 56 (1999) 271-316.  
In 1768 Robert Smith had enabled the Smith's Prize competition.  Barrow-Green writes: "During the eighteenth century the competition quickly became established as Cambridge’s premier mathematical contest, although its reputation was limited to the Cambridge mathematical community. To those outside Cambridge, victory in the mathematical Tripos was still the ultimate achievement. This confusion resulted from a lack of understanding of the nature of the two contests. Although both consisted of sets of examinations, the examinations were of a fundamentally different character.  On the one hand, the Tripos was a problem-solving marathon second to none. Its multitude of papers contained more questions than could be solved in the allotted time and success depended more on having the mechanical ability to solve problems as rapidly as possible than on having a clear understanding of the theory. On the other hand, the Smith’s Prize examination consisted of only a few papers and was generally aimed at soliciting a more thoughtful or philosophical approach to the questions asked. The level of questioning was of a higher standard and candidates were expected to show insights not required in the Tripos."
A: Returning to the original question about public mathematical contests, possibly Burton borrowed from Oystein Ore's Cardano, The Gambling Scholar (Princeton University Press 1953).  In "The Battles of the Scholars" (pages 53-107), Ore places the two events (1535 and 1548) in the context of the times.  He writes on page 99 that "It was a gala occasion in Milan when the dispute finally took place on August 10, 1548, in the Church in the Garden of the Frati Zoccolanti...  High officers, nobles, and distinguished citizens were present and Don Ferrante di Gonzaga, the governor of Milan, had been named the supreme arbiter.  The topic of the dispute was to be the problems..."
How very public that was!  Note, though, that Ore refers to the occasion as a dispute rather than a contest.  Throughout the chapter, Ore describes the acerbity accompanying mathematical disputes among 16-century Italian mathematicians, and it is well known that "disputations" were the predecessors of written dissertations in European universities. 
Nevertheless, Ore does use the word "contest" for the 1535 event, for which there were to be "thirty questions and the loser was to pay for a corresponding number of banquets for the winner and his friends."  (p. 63) 
The question remains, if I may tighten it by 200 years:  are there no other known examples of mathematical contests before 1600?
A: If you are willing to extend "contests" to "disputations" as your latest answer seems to suggest, what certainly should be mentioned is the drawn-out disputation over the indivisibles that turned particularly acerbic because of its connection to the eucharist and a perceived theological threat to the latter. Stefano degli Angeli wrote some spirited defenses of the indivisibles of his teacher Cavalieri against attacks emanating mainly from jesuit scholars, including a claim that they constituted such a threat to Euclidean geometry as to destroy it or be themselves destroyed (Tacquet). Shortly afterwards, degli Angeli's jesuat order was shut down by papal brief.
