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в начале века stresses
gen.	at the beginning of the century
polit.	in the early (in the early 1800s – в начале 19 века elmajor)
в начале XXI века
hist.	in the early 21st century (Alex_Odeychuk)
в начале XX века
hist.	at the outset of the 20th century (Alex_Odeychuk)
в начале XXI века
hist.	in the early twenty-first century (Alex_Odeychuk)
в начале XX века
Makarov.	in the early 20th century

в начале XXI века: 22 phrases in 6 subjects

American usage, not spelling	1
General	10
Historical	1
Makarov	7
Mathematics	2
Scientific	1