1. I won the first prize in the speech contest.

2. I won first prize in the speech contest.

(Do I have to use 'the' or not?)

3. You'll be making a business trip to Korea in an early July.

4. You'll be making a business trip to Korea in the early July.

(Do we have to use 'the ' or 'a' in #3 and #4?)

5. We are going to give/have/ throw a party.
(Are all verbs OK?)

6. Is Kelly a female name or a male name?

2 is better than 1.

4 and 5 -- don't use either "an" or "a" or "the"
You'll be making a business trip to Korea in early July.

5 -- all the verbs are OK.

6 -- as a first name, it's mostly a girl's name, but I've seen it for both.

1. Both sentences are grammatically correct, but the use of "the" in the first sentence implies that there was only one speech contest in which you won the first prize. In the second sentence, the absence of "the" suggests that you won the first prize in a particular speech contest, but there may have been other speech contests as well.

To determine which one to use, consider the context and whether there is a specific speech contest that you are referring to. If there is only one speech contest being discussed, use "the." If there might be multiple speech contests, use "a" or omit the article altogether.

3. The correct sentence would be: "You'll be making a business trip to Korea in early July."

In this case, "the" is not necessary because "early July" refers to a general time period and not a specific instance of it. When talking about a general time period, it is not necessary to use an article like "the."

4. "You'll be making a business trip to Korea in the early July" is not grammatically correct. It should be "You'll be making a business trip to Korea in early July" or "You'll be making a business trip to Korea in the early part of July," depending on the intended meaning.

Again, since "early July" refers to a general time period, the use of "the" is not required.

5. All three verbs, "give," "have," and "throw" can be used to describe organizing a party. However, there are slight differences in meaning:

- "Give a party": This implies that you are hosting the party and providing the location, food, and drinks.
- "Have a party": This can be used interchangeably with "give a party" and implies hosting and organizing the event.
- "Throw a party": This suggests a livelier and more energetic event, often involving music, dancing, and a larger group of people.

Choose the verb based on the specific tone and atmosphere you want to convey for the party.

6. Whether Kelly is a female or male name depends on the person in question. Kelly can be a name for both males and females. It can be used as a given name for girls or a surname for both genders. It's important to consider the context or ask the person in question if you want to know their gender.