arithmetic sequences - 2024 ACF Winter

Question

Salem–Spencer sets cannot contain any non-trivial examples of these sequences, which are the subject of Szemerédi's theorem. For 10 points each:

[10e] Name these sequences in which the difference between any two consecutive terms is constant, unlike in geometric sequences.

ANSWER: arithmetic sequences [or arithmetic progressions]

[10h] In 2004, Szemerédi's theorem was extended with this result, which states that the sequence of prime numbers contains arbitrarily long arithmetic sequences.

ANSWER: Green–Tao theorem

[10m] The Green–Tao theorem led to improved results concerning the first Hardy–Littlewood conjecture, which generalizes this conjecture. This conjecture has been verified up to 10-to-the-388342, somewhat larger than the 10-to-the-18 upper bound for Goldbach’s conjecture.

ANSWER: twin prime conjecture

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Summary


2024 ACF Winter at UC Berkeley	2024-11-16	Y	3	20.00	100%	33%	67%
2024 ACF Winter at Clemson	2024-11-16	Y	7	14.29	100%	14%	29%
2024 ACF Winter at Lehigh	2024-11-16	Y	7	15.71	100%	14%	43%
2024 ACF Winter at Northwestern	2024-11-16	Y	9	17.78	100%	33%	44%
2024 ACF Winter at Ohio State	2024-11-16	Y	6	13.33	100%	17%	17%
2024 ACF Winter at Online	2024-11-16	Y	7	12.86	100%	29%	0%
2024 ACF Winter at UBC	2024-11-16	Y	2	25.00	100%	100%	50%
2024 ACF Winter at Central Florida	2024-11-16	Y	5	12.00	100%	0%	20%
2024 ACF Winter at Oxford	2024-11-16	Y	10	14.00	100%	30%	10%

Data


Berkeley A	Stanford L	10	10	10	30
Berkeley B	Berkeley C	10	0	0	10
Stanford M	Stanford A	10	10	0	20
Alabama A	Auburn C	10	0	0	10
Auburn A	Clemson A	10	10	0	20
Emory B	Georgia Tech A	10	0	0	10
Georgia Tech B	Georgia A	10	10	0	20
Georgia Tech D	Georgia Tech C	10	0	0	10
Emory A	Georgia Tech F	10	0	0	10
Georgia Tech E	Tennesse A	10	0	10	20
Columbia A	Penn State B	10	10	0	20
Penn State A	Columbia B	10	0	0	10
Johns Hopkins A	Rowan A	10	10	0	20
Penn A	Johns Hopkins B	10	10	0	20
Princeton A	Lehigh A	10	0	0	10
Haverford A	Rutgers A	10	0	10	20
Bard	Rutgers C	10	0	0	10
Indiana B	WashU D	10	0	0	10
Purdue A	Miami	10	0	0	10
WashU B	Northwestern A	10	10	10	30
Notre Dame	UChicago C	10	0	0	10
Indiana A	Purdue C	10	10	0	20
UChicago B	Purdue D	10	0	10	20
UIUC A	SIUE A	10	10	0	20
UIUC C	UChicago D	10	0	10	20
UChicago A	WashU C	10	10	0	20
Pitt A	CWRU B (UG)	10	0	10	20
Kenyon A (UG)	Michigan D (DII)	10	0	0	10
Michigan A	Michigan C (UG)	10	10	0	20
Michigan B	Ohio State A (UG)	10	0	0	10
Michigan State A	CWRU A (UG)	10	0	0	10
CWRU C (UG)	Pitt B (UG)	10	0	0	10
Vassar B	Colorado College	10	0	0	10
Texas B	McGill E	10	0	10	20
Missouri	NYU B	10	0	0	10
NYU A	Vassar A	10	0	0	10
Texas A	Mississippi State	10	0	10	20
Texas C	Oregon State	10	0	0	10
WUSTL A	Iowa	10	0	0	10
SFU	UW B	10	10	10	30
UW A	UBC B	10	0	10	20
Florida Tech A	Florida Tech B	10	0	0	10
UF E	UCF B	10	0	0	10
UF B	UF C	10	0	0	10
UF A	UF D	10	10	0	20
UCF A	UF F	10	0	0	10
Durham B	Bristol B	10	0	0	10
Bristol A	Cambridge C	10	0	0	10
Durham A	Imperial B	10	0	0	10
LSE A	Cambridge B	10	0	0	10
Manchester	Cambridge A	10	0	0	10
Oxford A	Imperial A	10	10	10	30
Oxford B	Cambridge D	10	0	0	10
Edinburgh	Vanderbilt	10	0	10	20
Birmingham	Warwick A	10	0	10	20
LSE B	Warwick B	10	0	0	10