Auto comment: topic has been updated by Bovmelo (previous revision, new revision, compare).

"consider"

"It can be shown that"

It is obvious that

Trust me bro

Let's assume

I hate it when they use "it is easy to see that"

We can notice that

“bitset”

well known trick in china!

Hence Proved !

It can be proved that...

somehow "it works" :)

Pretest Passed or Accepted in Contest

We can achieve this by. ..

" We don't have the strict proof, but we have faith and stress."

The proof is left as an exercise to the reader.

Without loss of generality

let it be

TLE

Well-Known Fact

obviously

This is a well-known fact that

It can be shown that there is always at least one answer. I have seen it dozens of times

I "Claim"

"I am lgm"

"Observation:"

from "someoldname" 's theorem it's obvious that...

"There are other approaches..."

We can prove by induction that ...

"from the following observations"

If you actually want to prove something: "Assume that"

Once you realize that you don't know how to prove something: "It's obvious that"

The proof is left as homework.

Source : https://codeforces.net/blog/entry/9070 (word "homework" written in tutorial of 351C + 351D)

"Time complexity: O( )"

"Maximum Value among the minimums"

It can be proven that

We can use (smth) to optimize $$$\mathcal{O}(n^2)$$$ to $$$\mathcal{O}(n \log n)$$$ due to monge property

(How?)

"First of all"

Editorial

Let's start with a naive solution

"We are programmers not mathematicians"

It was revealed to me in a dream