ὁ ἀγών, τοῦ ἀγῶνος, contest, struggle; trial   As we can see, the stem has an omega. How do we know? Take the genitive form (always provided in the dictionary, together with the word itself and the article to indicate gender), subtract the ending -ος and we are left with the stem (We are talking 3rd declension of course):   παίς, παιδός, ὁ child, boy ὄνομα, ὀνόματος, τό name ἀγών, ἀγῶνος, ὁ contest, struggle   Remember the basic 3rd declension endings, which will be added to the stem of the noun.

• In the nominative and vocative singular the omega sound is shortish, as the acute accent tells us. Actually, it's a question of pitch, of voice only going up /, not up / and then down \ again as in /\ (using circumflex to show the up and down movement of pitch):
    τοῦ χειμῶνος  or
    τῶν ἀγρῶν but:
    ὁ ἀγών, ὁ χειμών 
• Genitive plural ending  ΩΝ  is always long, so the syllable before, if stressed, has got to be shortish, so acute accent.
• all the other omegas are really long and have to wear a circumflex to show it.
• Dative plural always ends in  ΣΙ . But the Greeks don't like
  νσι 
  δσι 
  τσι 
  so they shorten the ending by dropping the offending
   ν:    τοῖς χειμῶ σι 
   δ:    τοῖς παι σί 
   τ:    τοῖς ὀνόμα σι 

• lesson 4a: There is going to be a festival in Athens, and the ladies would like to go and see the contests, τοὺς ἀγῶνας 
• the cold  τοῦ χειμῶνος  of winter
• the force τῶν χειμώνων  of storms
• οἱ ἀγῶνες are spectacular
• θαυμάζομεν οὖν αὐτούς, τοὺς ἀγῶνας 
• We freeze  ἐν τῶ χειμῶνι  in winter
• sailors used to be shipwrecked a lot in storms: ἐν τοῖς χειμῶσι  
• ὦ χειμών , why are you so bitterly cold?
• παῦε , ὦ χειμών ...  stop, you ... [censured] storm!

αἰών, αἰῶνος, ὁ, space of time; age, lifetime; one's destiny
χιτών,χιτῶνος, ὁ, a tunic, undergarnment
δελφίς, δελφῖνος, ὁ, a dolphin
ἱδρώς, ἱδρῶτος, ὁ, sweat, perspiration