# Tagged Questions

The function that maps a real number $x$ to the smallest integer greater than or equal to $x$ (which is often denoted $\lceil x\rceil$. See also (floor-function).

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### Relation between a floor and a ceiling function for a problem

I was trying to formulate some problem. I want to find a relation between a floor and ceiling function. Suppose the Property 1 satisfies that it has $\lfloor \frac{n}{2} \rfloor$ number of $X$. Then ...
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### Find when ceiling functions differ by one

For the function $$f(x) = (a/b)x + (c/b)$$ how do I find the smallest value x such that: $$\lceil f(x)\rceil < \lceil f(x) + kx\rceil$$ where x is a positive integer, a, b, and c are ...
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### Integrating $\int^b_a [x]\,dx+\int^b_a [-x]\,dx$

I came across a question today... Integrate $\int^b_a [x]\,dx+\int^b_a [-x]\,dx$ where [.] denotes greatest integer function is equal to Now this question is not helpful for me because in that ...
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### How to solve mixed-integer problem?

I don't know how to solve this equation, $$\left\lceil\frac{x-A}{B}\right\rceil C + D x < E, \quad x\in \mathbb{Z}$$ In this equation, only $x$ is unknown and $x$ is integer, but $A,B,C,D,E$ are ...
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### Floor-Ceil Properties

If $n$ and $k$ are integers, with $k$ different from zero: $$\left\lceil{\frac{n+1}{k}}\right\rceil = \left\lfloor{\frac{n}{k}}\right\rfloor + 1$$ How can I prove this property? I would appreciate ...
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### Floor and ceiling opposite property

For $x\in \mathbb{R}$ let's define $[x]$ as: $$[x] = max \{ k\in \mathbb{Z}: k\leq x \}$$ and $[x]^{*}$ as: $$[x]^{*} = min \{ k\in \mathbb{Z}: k\geq x \}.$$ Show that: $$[x]^{*} = -[-x].$$ So ...
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### Summation of $\sum_{k=1}^{n}\left \lfloor \log _{m}k \right \rfloor$ and $\sum_{k=1}^{n}\left \lceil \log_{m}k\right \rceil$ [closed]

$$\sum_{k=1}^{n}\left \lfloor \log _{m}k \right \rfloor$$ $$\sum_{k=1}^{n}\left \lceil \log_{m}k\right \rceil$$ I found myself stuck trying to solve these two summations but i can't make any progress....
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### ceiling functions inequality

Please, help me in solving this ceiling function inequality. $\lceil n/4 \rceil \ge 3$ I know the formal definiton of the ceiling functions: $\lceil x \rceil = n$ iff $n-1< x \le n$ ...
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### Possible circular reasoning in textbook proof that $\lceil x+m\rceil=\lceil x\rceil +m$

The goal is to prove that $\lceil x+m\rceil=\lceil x\rceil +m$, where $x$ is a real number and $m$ is an integer. The book outlines the following proof: Write $x=n-\epsilon$, where $n$ is an ...
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### How to pigeonhole the primes between $p_n$ and $p_{n+1}^2$ for twin prime conjecture?

For any full list of the primes up to the $n$th prime: $P = \{2, 3,5,\dots, p_n\}$, any natural number $q$ such that $p_n \lt q \lt p_{n+1}^2$ that is not sieved by a prime in $P$ is also a prime. ...
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### Find an expression for the ones digit of a positive integer using a floor or ceiling function.

I am trying to find out how to write an expression for the ones digit for any given positive integer. For example, if n = 326, the expression should evaluate to 6. The only things I've been able to ...
I would like to know if there is a way to express the "roundup" function of excel or "rounding off of a number to the nearest whole number" in an equation form. e.g. in excel: roundup $(2.13,0) = 3$, ...
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