Confusion in Generalization of Pigeonhole Principle

Question

I want to ask if this two statements are correct:

1. If N objects are placed into k boxes, then there is at least one box containing at most ⌈N/k⌉ objects. (originally the statement is "at least" ⌈N/k⌉ objects).

and similarly

2)If N objects are placed into k boxes, then there is at least one box containing at least ⌊N/k⌋ objects. (originally the statement is "at most" ⌊N/k⌋ objects).

Intuitively, those two statements are correct since each box will get ⌊N/k⌋ object at minimum, and if there is a reminder some of the boxes will get ⌈N/k⌉ objects. I am confused how should I use the generalizations. What do I miss about differences? Are my edited versions correct ?

for proof of statement1:

proof by contradiction: assume all boxes have ⌈N/k⌉ + 1 objects,

(⌈N/k⌉ + 1)*k > N/k * k = N (contradiction) .

It seems correct but Idk how to interpret this

Answer 1

It sounds like you have a misunderstanding of what the Pigeonhole Principle says: It actually says that some box has at least so many items no matter how you distribute the items in the boxes. It definitely is not limited only to attempts to distribute the items evenly in the boxes (and if it were it would be much less useful).