free product - 2025 NAFTA

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%

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Conversion

Team	Opponent	Part 1	Part 2	Part 3	Total	Parts
(give me an) ANSWER: die for baseball	Quebecois Olga Tokarczuk be like 'freins de jacob interdits'	0	10	0	10	E
1000000 Delinquents	Invective Against Swans	0	0	10	10	M
A Neg of Beauty is a Joy Forever	Vacuous Tortoise	0	10	10	20	EM
BHSU	In the Mood for Transportation Equipment	0	10	10	20	EM
Berkeley B	Berkeley A	0	10	10	20	EM
Capital One in the Twenty-First Century	consumption smoothing by borrowing future big law income	0	10	10	20	EM
Chicago !A	Muscle Mass Extinction Event	0	10	10	20	EM
Constans Constantius and Constantine Jr.	Not Old! (Old)	10	10	10	30	HEM
Dour Straits	The Third Time when she was Given to Answer the Hard and the Easy, and Answered Only the Easy	0	10	10	20	EM
Execute Order 6 7	Northwestern	10	10	10	30	HEM
H Charlson's Washed Up Scrubs	C R A B S	0	10	10	20	EM
Habitat 67	NYU B or not NYU B (that is the question)	0	10	10	20	EM
I fear not the man who has reviewed 10000 cards once, but I fear the man who has reviewed one card 10000 times	Licence to Bill	0	10	0	10	E
Indiana	The tram is not in service; please use the moving walkway.	0	10	10	20	EM
It's something unpredictable/that in the end is right/please refrain from e-mailing my wife	David Foster Gromit	0	10	10	20	EM
JHU	Maryland F♯ A♯ ∞	0	10	10	20	EM
Middlesex County Fetty Wap Fan Club	Surrealist Subcommandante Marcos be like Esto no es una pipa	0	10	0	10	E
No Country for Old Quizzers	Dielectrical Materialism	0	10	10	20	EM
Penn State	Munir Siddiqui Fan Club	0	10	10	20	EM
Polish Orthographers Posting Their Łs	Copán Seethe	0	10	10	20	EM
Shipping Down from Ballston	NYU A	0	10	0	10	E
Six Quizbowlers in Search of Owen Minmo	Negged Five Times in a Yellow Cab Against the Gottis and Gambinos	0	10	0	10	E
Stanford A	Stanford B	0	10	0	10	E
Strange Case of Dr. Joker and Mr. Pennywise	Guelph Cloze	0	10	0	10	E
Team 8	Team 7	0	10	0	10	E
The Saint-Straw that Broke the Camille's Back	The Fire Neg Time	0	10	10	20	EM
The Unbearable Heaviness of History Players	UBC B	0	10	10	20	EM
The lion does not concern itself with negging teammates out of their own categories	Licence to C. Wright Mill	0	10	0	10	E
UBC A	Abbas I founded the Abbasids	0	10	0	10	E
UNC Charity	Rosslyn Academic Team	0	10	0	10	E
Untitled Keyal Team	Noah Sheidlower	0	10	10	20	EM
Untitled by Alexander McQueen by Premeditated Chaos by Alexander McQueen by Percival Everett	The "C" Team	10	10	10	30	HEM
William of Baskerville attempts to investigate John P. Robarts, gets turned away for not having a student card	Where's the nearest library, I need to have a public crashout.	0	10	10	20	EM
You Don't Have the Cards	I'm just here (being trivia club president) so I won't get fined	0	10	0	10	E
https://i.imgur.com/C6nN2FE.png	Contessa Dora	0	10	10	20	EM

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%