New posts in pigeonhole-principle

Are there rigorous formulation and proof of the pigeonhole principle?

if there are 5 points on a sphere then 4 of them belong to a half-sphere.

Chess Master Problem

Prove using induction that from a set of $n+1$ numbers from $\{1,2,..2n\}$, at least one number will evenly divide another.

Prove that at a party of $25$ people there is one person knows at least twelve people.

The pigeonhole principle and a professor who knows $9$ jokes and tells $3$ jokes per lecture

Using Pigeonhole Principle to prove two numbers in a subset of $[2n]$ divide each other

Of any 52 integers, two can be found whose difference of squares is divisible by 100

A discrete math riddle

7 fishermen caught exactly 100 fish and no two had caught the same number of fish. Then there are three who have together captured at least 50 fish.

100 Soldiers riddle

What is your favorite application of the Pigeonhole Principle?

A natural number multiplied by some integer results in a number with only ones and zeros

For an irrational number $\alpha$, prove that the set $\{a+b\alpha: a,b\in \mathbb{Z}\}$ is dense in $\mathbb R$ [closed]