I I want to tell someone to make /do competition with me or with other guy, what is the right way to express it?

1. Let's do/ make competition!

2. Let do competition!

3. Let make competition!

4. Let's do competition!

Please, I would like to understand the using of 's after let (let's), if any. Also, I would like to understand the distinction between 'make' and 'do'. Thank you

If I want to tell someone to make /do competition with me or with other guy, what is the right way to express it?

1. Let's do/ make competition!

2. Let do competition!

3. Let make competition!

4. Let's do competition!

I would like to understand the using of 's after let (let's), if any. Also, I would like to understand the distinction between 'make' and 'do'.