Question
These sequences can be used to prove Hurwitz’s (“HUR-vitz’s”) theorem concerning rational approximation. For 10 points each:
[10h] Name these sequences that can also be used to show that Ford circles are tangent to each other. The nth one of these sequences is the set of all reduced fractions with denominators less than or equal to n.
ANSWER: Farey sequences [accept Farey series]
[10m] The length of the nth Farey sequence equals this function of n plus the length of the previous sequence. This function of n equals n times the product over p of the expression: “one minus one over p,” where p is a prime factor of n.
ANSWER: Euler’s (“OY-lurz”) totient (“TOH-shint”) function [or Euler’s phi function]
[10e] The totient function is used to generalize this French mathematician’s “little theorem,” which is useful when performing modular arithmetic.
ANSWER: Pierre de Fermat (“FER-mah”) [accept Fermat’s little theorem]
Data
| Team | Opponent | Part 1 | Part 2 | Part 3 | Total |
|---|---|---|---|---|---|
| Brown A | Cornell A | 10 | 0 | 10 | 20 |
| Duke A | Florida A | 0 | 0 | 10 | 10 |
| Florida B | Columbia B | 0 | 0 | 10 | 10 |
| Georgia Tech B | Iowa State A | 0 | 10 | 10 | 20 |
| Illinois A | McGill A | 0 | 0 | 10 | 10 |
| Imperial A | Minnesota B | 0 | 0 | 10 | 10 |
| Indiana A | Chicago C | 0 | 0 | 10 | 10 |
| Johns Hopkins A | Purdue A | 0 | 0 | 10 | 10 |
| Maryland A | Ohio State A | 0 | 10 | 10 | 20 |
| Minnesota A | Columbia A | 0 | 0 | 10 | 10 |
| North Carolina A | Toronto A | 0 | 0 | 10 | 10 |
| Northwestern A | Harvard A | 0 | 0 | 10 | 10 |
| Rutgers A | Claremont A | 0 | 0 | 10 | 10 |
| Rutgers B | Michigan A | 0 | 0 | 10 | 10 |
| South Carolina A | Penn State A | 0 | 0 | 10 | 10 |
| Stanford A | Chicago A | 0 | 10 | 10 | 20 |
| Texas A | MIT A | 0 | 10 | 10 | 20 |
| UC Berkeley A | Cornell B | 10 | 10 | 10 | 30 |
| UC Berkeley B | Houston A | 0 | 10 | 10 | 20 |
| Vanderbilt A | NYU A | 0 | 0 | 10 | 10 |
| WUSTL A | Chicago B | 0 | 0 | 10 | 10 |
| WUSTL B | Penn A | 0 | 10 | 10 | 20 |
| Yale A | Georgia Tech A | 0 | 10 | 10 | 20 |
| Yale B | Virginia A | 0 | 0 | 10 | 10 |