Floor and Ceiling Functions

Ceiling Function


Which of the following statements is/are true?

I. x+n=x+n\lceil x+n\rceil=\lceil x\rceil+n for any real xx and any integer n.n.
II. x+x=1\lceil x\rceil+\lceil-x\rceil=1 if and only if xx is not an integer.
III. x+yx+yx+y+1\lceil x\rceil+\lceil y\rceil\le\lceil x+y\rceil\le\lceil x\rceil+\lceil y\rceil+1 for any real xx and y.y.

What is the solution set to x38=10?\lceil\frac{x}{3}-8\rceil=10?

What is log559?\lceil\log_{5}59\rceil?

How many integers xx satisfy the equation log3x=4?\lceil\log_{3}x\rceil=4?

What is π?\lceil-\pi\rceil?


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