Packet 5: Bonus 5
This operation is applied “with amalgamation” to the fundamental groups of two overlapping, path-connected spaces in the Seifert-Van Kampen theorem. For 10 points each:
[10h] Name this binary operation on two groups, which outputs the group consisting of all words that do not contain consecutive elements from the same group or superfluous uses of the identity.
ANSWER: free product [accept amalgamated free product or free product with amalgamation; prompt on product]
[10e] The free group on n generators is formed via the free product of this group with itself n times. This group is, up to isomorphism, the unique infinite cyclic group and is denoted with a blackboard capital Z.
ANSWER: integers [accept descriptions of the integers under addition]
[10m] The proof of this result uses the fact that the group of rotations in R3 (“R-three”) has a free subgroup on two generators. This result states that any two balls of different volumes are equidecomposable by isometries.
ANSWER: Banach-Tarski paradox
<FW, Other Science (Math)> | NAFTA-Packet-5
| Heard | PPB | E % | M % | H % |
|---|---|---|---|---|
| 35 | 17.14 | 97% | 66% | 9% |
Conversion
| Team | Opponent | Part 1 | Part 2 | Part 3 | Total | Parts |
|---|---|---|---|---|---|---|
| Berkeley B | Berkeley A | 0 | 10 | 10 | 20 | EM |
| Constans Constantius and Constantine Jr. | Not Old! (Old) | 10 | 10 | 10 | 30 | HEM |
| Stanford A | Stanford B | 0 | 10 | 0 | 10 | E |
| Team 8 | Team 7 | 0 | 10 | 0 | 10 | E |
Summary
| Tournament | Edition | Match | Heard | PPB | E % | M % | H % |
|---|---|---|---|---|---|---|---|
| 2026 NAFTA at Stanford | 01/17/2026 | ✓ | 4 | 17.50 | 100% | 50% | 25% |
| 2026 NAFTA at UBC | 01/17/2026 | ✓ | 2 | 15.00 | 100% | 50% | 0% |
| 2025 NAFTA Online | 02/14/2026 | ✕ | 4 | 17.50 | 75% | 100% | 0% |
| 2026 NAFTA at Vanderbilt | 02/14/2026 | ✕ | 3 | 16.67 | 100% | 67% | 0% |
| 2025 NAFTA at Toronto | 09/13/2025 | ✓ | 5 | 12.00 | 100% | 20% | 0% |
| 2025 NAFTA at Maryland | 09/27/2025 | ✓ | 4 | 17.50 | 100% | 75% | 0% |
| 2025 NAFTA at Harvard | 10/04/2025 | ✓ | 3 | 16.67 | 100% | 67% | 0% |
| 2025 NAFTA at Oxford | 10/11/2025 | ✓ | 4 | 20.00 | 100% | 100% | 0% |
| 2025 NAFTA at Chicago | 11/08/2025 | ✓ | 6 | 18.33 | 83% | 83% | 17% |
| 2025 NAFTA at Columbia | 11/08/2025 | ✓ | 5 | 18.00 | 100% | 60% | 20% |
| 2025 NAFTA at Richmond | 12/20/2025 | ✓ | 2 | 20.00 | 100% | 100% | 0% |