power series - 2025 ACF Regionals

Question

Answer the following about the construction of the p-adic numbers, for 10 points each.

[10e] The p-adic numbers can be formally defined as one of these constructs expressed using a rational times all powers of p. Taylor and Maclaurin name examples of these expansions used to represent smooth functions.

ANSWER: power series [accept formal series; prompt on infinite summation]

[10m] Under the p-adic norm, the p-adics are a completion of the rationals, meaning that each of these sequences converges to a limit in the set. In one of these sequences, all terms past a certain point become arbitrarily close.

ANSWER: Cauchy (“KO-shee”) sequences

[10h] To obtain other completions in the field of fractions, one can localize a type of integral domain named for this mathematician. A construction named for this mathematician partitions the rationals into “left” and “right” sets.

ANSWER: Richard Dedekind [accept Dedekind domain or Dedekind cut]

<AR, Other Science>

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Summary


California	2025-02-01	Y	3	16.67	100%	33%	33%
Florida	2025-02-01	Y	3	13.33	67%	33%	33%
Lower Mid-Atlantic	2025-02-01	Y	6	15.00	100%	17%	33%
Midwest	2025-02-01	Y	6	18.33	100%	33%	50%
North	2025-02-01	Y	3	20.00	100%	33%	67%
Northeast	2025-02-01	Y	5	22.00	100%	60%	60%
Overflow	2025-02-01	Y	5	14.00	100%	20%	20%
South Central	2025-02-01	Y	2	20.00	100%	50%	50%
Southeast	2025-02-01	Y	4	12.50	75%	0%	50%
UK	2025-02-01	Y	10	21.00	100%	70%	40%
Upper Mid-Atlantic	2025-02-01	Y	8	18.75	100%	25%	63%
Upstate NY	2025-02-01	Y	3	16.67	100%	33%	33%

Data


Claremont B	Claremont A	10	0	0	10
UC Berkeley A	UC Berkeley B	10	10	10	30
UCSD	UCLA	10	0	0	10
Valencia A	Florida B	10	0	0	10
Florida A	Florida State A	10	10	10	30
UCF B	UCF C	0	0	0	0
Duke	Virginia Tech A	10	0	10	20
Wake Forest	Liberty A	10	0	0	10
UNC B	Liberty B	10	10	10	30
Virginia A	South Carolina	10	0	0	10
UNC C	Liberty C	10	0	0	10
UNC D	William & Mary	10	0	0	10
Illinois A	Chicago A	10	10	10	30
Illinois B	Chicago D	10	0	0	10
Chicago C	Illinois C	10	0	10	20
Indiana A	Indiana B	10	10	0	20
WashU B	Missouri	10	0	0	10
WashU A	Missoui S&T	10	0	10	20
Carleton	Minnesota B	10	0	0	10
Iowa State	Wisconsin A	10	10	10	30
Winona State	Wisconsin B	10	0	10	20
BU	UMass Boston	10	0	0	10
Harvard B	Brandeis B	10	10	10	30
Brown A	Dartmouth A	10	10	10	30
Tufts A	MIT A	10	0	0	10
Harvard A	Brandeis A	10	10	10	30
Purdue	Georgetown B	10	0	0	10
Georgetown A	Iowa	10	0	0	10
Notre Dame C	Northwestern A	10	0	0	10
Northwestern B	Notre Dame B	10	10	10	30
Stanford B	Notre Dame A	10	0	0	10
TAMU	ASU	10	0	0	10
Texas A	Texas B	10	10	10	30
Louisville A	Alabama A	10	0	0	10
Georgia Tech C	Georgia Tech A	10	0	10	20
Vanderbilt A	Tennessee A	0	0	0	0
Bruin A	Georgia Tech D	10	0	10	20
Cambridge A	Cambridge E	10	10	10	30
Cambridge B	Warwick B	10	10	10	30
Warwick A	Cambridge C	10	10	0	20
Bristol	Cambridge D	10	10	10	30
Durham	Oxford B	10	10	0	20
NYU C	Imperial A	10	0	0	10
LSE	Oxford C	10	10	0	20
Sheffield	Imperial B	10	0	0	10
Oxford A	Southampton A	10	10	10	30
Manchester	Southampton B	10	0	0	10
NYU A	Columbia B	10	0	10	20
Columbia C	Haverford A	10	0	10	20
John Jay College	Maryland A	10	0	10	20
Johns Hopkins A	Vassar A	10	10	10	30
Maryland B	Penn A	10	0	0	10
Yale A	NYU B	10	0	0	10
Penn B	George Washington B	10	0	0	10
Rutgers A	George Washington A	10	10	10	30
Binghamton A	Cornell D	10	0	0	10
RIT A	ESF A	10	10	10	30
RIT B	Cornell C	10	0	0	10