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"La argentina" is a form of "argentino", a noun which is often translated as "Argentinian". "Perú" is a proper noun which is often translated as "Peru". Learn more about the difference between "Perú" and "la argentina" below.
el argentino, la argentina, argentino(
ahr
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hehn
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tee
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noh
)This means that the noun can be masculine or feminine, depending on the gender of the noun it refers to (e.g., el doctor, la doctora).
a. Argentinian 
Unos argentinos que conozco tienen un bar en el centro.Some Argentinians I know own a bar in the city center.
b. Argentinean  (United States)
Regionalism used in the United States
Daniela es una argentina que vive en Madrid.Daniela is an Argentinean living in Madrid
c. Argentine 
El coordinador del posgrado es un argentino.The postgraduate coordinator is an Argentine.
An adjective is a word that describes a noun (e.g., the big dog).
b. Argentinean  (United States)
Regionalism used in the United States
Las empanadas argentinas son de mis comidas favoritas.Argentinean pasties are one of my favorite foods.
c. Argentine 
El gobierno argentino emitirá billetes de mayor denominación.The Argentine government will issue higher denomination notes.
a. silvery 
Este pintor aprecia mucho la luz argentina de la madrugada.This painter loves the silvery light of the early morning.
b. argentine  (literary)
A word or phrase restricted in usage to literature or established writing (e.g., sex, once upon a time).
Parecía un hada con su pelo argentino, su vestido largo y sus joyas brillantes.She looked like a fairy with her argentine hair, long dress, and glittering jewels.
4. (figurative) (tinkling)
A phrase used as a figure of speech or a word that is symbolic in meaning; metaphorical (e.g., carrot, bean).
a. silvery 
Me alegra escuchar la risa argentina de Mónica.It's good to hear Monica's silvery laughter.