A bBirthday, a "P" and the aAnswer to eEverything (fFirst pPart)

Proper capitalization in your title.

Yesterday I went to a friend of mine'’s birthday party.

“Friend of mine’s” is awkward

I might have mentioned it in one of my texts here on LangcCorrect.

Capitalize “LangCorrect” consistently.

Even though the room where the language exchange takes place was closed, my friend also invited some people thatwho work at the organiszation that manages this event, so we could party there.

Use “who” instead of “that” for people.

everyone had to get dressed as something that startsed with the same first letter as their name.

Past tense “started” matches the invitation context.

Since my name is Pietro, my letter was "P"."

Place the period inside the quotation marks.

Initially, I thought of some unoriginal costumes (like "the pirate"), but then I had an idea:

Add a comma after “Initially.”

I'm going to would dress up as a "problem"!

“I would” to match past tense narration.

Then I did the same thing withon the back of the image, where I wrote the solution: 42, a reference to the book “The Hitchhiker's Guide to the Galaxy”.”

Use “on the back” instead of “with the back.”

I wrote sentences like "It seems easy, but it's not" or "We explained this in class"."

Yesterday I went to a friend of mine's birthday party.

Your sentence is grammatically correct, but my correction is simpler and natural

everyone had to get dressed as something that starts with the same first letter asof their name.

Initially I thought of some unoriginal costumes (like "thea pirate"), but then I had an idea:

I might have mentioned it in one of my texposts here on Langcorrect.

There are a few specific things we use "text" to refer to, it's kind of complex to explain but it tends to focus on the aspect of it being a bunch of character strings and not so much the content. ie. "copy and paste the text," "highlight the text," note that in this usage "text" is always singular.
We also call typed messages sent through phone service "texts," and we call books "texts" but that sounds kind of academic/archaic
It doesn't sound natural here.

Even though the room where the language exchange takes place was closed, my friend also invited some people that work at the organisation that manages thise event, so we couldand they allowed us to party there.

This sentence is kind of confusing without clarifying that they gave you access to the room

Since my name is Pietro, my letter was "P"."

commas and periods go inside quotes

Initially I thought of some unoriginal costumes (like "thea pirate"), but then I had an idea:

I wrote a funny and meaningless math formula and I generated with ChatGPTused ChatGPT to generate a math frame.

I even bought some post-its whereon which I wrote some funny sentences to "help" people solve the problem.

sounds like you bought post-its at the location where you wrote funny sentences