It should be din, I think: http://forvo.com/word/%D0%B4%D0%B5%D0%BD%D1%8C/#ru
Добрый is a masculine adjective in the nominative case; доброе is a neutral adjective in the nominative case; доброго is a masculine adjective in the genetive case.
In Russian both masculine and feminine nouns can end with ь (soft sign). The best way to determine the gender of a word is to look it up in the словарь (dictionary-m.) These are a few ways you can determine if a noun ending in -ь is masculine: - the nouns are denoting male persons: учитель - teacher; вратарь - goalkeeper; парень - guy, fellow; король - king - months: январь - January; февраль - February; июнь - June - nouns ending in -тель: выключатель - switch, button switch; двигатель - engine, motor There are also a few ways to determine if a noun ending in -ь is feminine: - nouns denoting female persons like мать - mother and дочь - daughter - nouns ending in -жь, -шь, -чь, -щь: ночь - night; рожь - rye; помощь - help; брошь - brooch - nouns ending in -ость, -есть, like молодость - youth and свежесть - freshness But for instance площадь (area, square) is feminine, and so is тетрадь (notebook). So always use a dictionary! I hope this is helpful!
Can anyone elaborate on the spelling rule behind добрые? I guessed "добри"
Russian doesn't have a separate phrase for "good afternoon" (or a single word translation for "afternoon" for that matter), and English rarely uses "good day" as a greeting (it happens, but not often). So it makes sense to treat "добрый день" is the closest equivalent of "good afternoon".