# Questions tagged [ceiling-and-floor-functions]

This tag is for questions involving the greatest integer function (or the floor function) and the least integer function (or the ceiling function).

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### $\sum_{i=0}^k \lfloor\sqrt{ip} \rfloor = \frac{(p^2-1)}{12}$ where p is a prime and $p=4k+1$

Question: Let p be a prime number of the form 4k+1. Show that $\sum_{i=0}^k \lfloor\sqrt{ip} \rfloor = \frac{(p^2-1)}{12}$ Source: I came across this question while solving An Introduction to the ...
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### Simplifying expression with ceil

How can I simplify the following? $\lceil \frac{x}{\lceil \frac{x}{y} \rceil}\rceil$ I'm dealing with partitioning data for processing, where x is the total input size and y is the maximum chunk size ...
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### Find solutions for integrals with floor function

Find solutions of x satisfying: $$\int_{0}^{2\lfloor{x+14}\rfloor}\left(\frac{t}{2}-\left\lfloor{\frac{t}{2}}\right\rfloor\right)dt=\int_{0}^{x-\lfloor{x}\rfloor}\lfloor{t+14}\rfloor dt$$ My solution ...
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### Find $\{u \geq a\}$ for all $a \in \mathbb{R}$, $u(x)=\lfloor x\rfloor$

Let $u: \mathbb{R} \rightarrow \mathbb{R}$, $u(x)=\lfloor x\rfloor$. (i) Determine $\{u \geq a\}$ for all $a \in \mathbb{R}$. (ii) Show that $u$ is $\mathcal{B}(\mathbb{R}) / \mathcal{B}(\mathbb{R})$-...
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### How to integrate : $\int_{-1}^{\frac{3}{2}} \left[x^2\right] dx$ where [ ] represents Greatest Integer Function

Please explain to me this question. How to integrate: $$\int_{-1}^{\frac{3}{2}} \left[x^2\right] dx$$ where $[ \ \ ]$ represents Greatest Integer Function? I am not able to understand how do come to ...
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### Checking if there is a better way of handling the distribution of $K\times\lfloor \frac{X}{K}\rfloor$ given $X$ is already known.

Suppose we have a random variable $X$ and we know its distribution. How would we determine the distribution of $Y:=K\times\lfloor \frac{X}{K}\rfloor$ in relation to $X$, where $K$ is just any constant?...
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### residues mod. n of $\lfloor k\alpha\rfloor$ where $\alpha$ is irrational positive.

Consider for any integer $k\geqslant1$ : $$u_k=\lfloor k\sqrt2\rfloor$$ It's not difficult to prove that there are infinitely many $k$ such that $u_k$ is even and infinitely many $k$ such that $u_k$ ...
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### Given $x^5-x^3+x-2=0$, find $\lfloor x^6\rfloor$. [duplicate]

If $\alpha$ is a real root of the equation $x^5-x^3+x-2=0$, find the value of $\lfloor\alpha^6\rfloor$, where $\lfloor x\rfloor$ is the least positive integer not exceeding $x$. My approach is to ...
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### Simplify: $x\leq\lfloor\frac{2(y+2x)+1}{5}\rfloor$

I want to simplify this term: $$x\leq\left\lfloor\frac{2(y+2x)+1}{5}\right\rfloor$$ Any help would be most appreciated.
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### $Z$-transform of floor($\frac{n}{5}$) [closed]

I found problem about floor array that I can't solve. Find Z transform of f(n) = floor(n/5). I tried writing this array and for n from 1 to infinity, I got n/5 = 1/5 + 2/5 + 3/5 + 4/5 + 5/5 + 6/5... ...
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### Integrating floor functions without known limits

Suppose we have$$\frac{\displaystyle\int_0^n{\lfloor x \rfloor}\,dx}{\displaystyle\int_0^n{\lbrace x \rbrace }\,dx}$$ $n \in I$ where $\lfloor \cdot\rfloor$ and $\lbrace \cdot\rbrace$ represent the ...
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### Function missing squares. [duplicate]

Source: Taken from the math-olympiad textbook of A.Engel : Show that $$\bigg \lfloor{n+ \sqrt{n}+ \frac{1}{2}}\bigg \rfloor$$ is never equal to a square.
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### Proof: $\lfloor\sqrt{a_1}\rfloor+\lfloor\sqrt{a_2}\rfloor+ \dots+\lfloor\sqrt{a_{25}}\rfloor\ge\lfloor\sqrt{a_1+a_2+\dots+a_{25}+200k}\rfloor$ [duplicate]

Source: Proof: $[\sqrt {{a_1}} ] + [\sqrt {{a_2}} ] + \dots + [\sqrt {{a_{25}}} ] \ge \left[ {\sqrt {{a_1} + {a_2} + \cdots + {a_{25}} + 200k} } \right].$ Let $a_1,a_2,\dots,a_{25}$ be nonnegative ...
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### Showing that $S_n -\lfloor S_n \rfloor \sim U[0,1]$

$k \in \mathbb{N}$ is fixed $(X_n)_{n \geq 1}$ are all independent and follow an uniform law on $[0,k]$ We define $f(x)=x -\lfloor x \rfloor$ $S_n= \sum_{i=1}^{n} X_i$ $Z_n= f(S_n)$ We want to show ...
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### Find all $x$ for which $\left| x - \left| x-1 \right| \right| = \lfloor x \rfloor.$ [closed]

Find all $x$ for which $$\left| x - \left| x-1 \right| \right| = \lfloor x \rfloor.$$Express your answer in interval notation. I started by looking at x < 1 and x $\geq$ 1 separately, but I got ...
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### rounding numbers to 10ths, 100ths or 1000ths

I was going through the C++ 10th edition deitel's book. There I encountered a questioned regarding floor() function. And here's the question: Floor() practice question Now, from what I understand so ...
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### Find the number of points of discontinuity for $[\cot^{-1} x]$ [.] is the floor function

The range of $\cot ^{-1}$ is $[-\frac{\pi}{2},\frac{\pi}{2}]$ So the range for $[\cot ^{-1} ]$ is $\{-2,-1,0,1\}$ So there must be 4 points of discontinuity, but the answer says there are only three. ...
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### How to find range of $\left[\frac{[x]}{x}\right]$

If $[.]$ denotes greatest integer function, find the range of $$\left[\frac{[x]}{x}\right]$$ My friend and I tried solving this question and arrived at the answer ${0,1}$ but when we took the value ...
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### How to prove that for $a_{n+1}=\frac{a_n}{n} + \frac{n}{a_n}$ , we have $\lfloor a_n^2 \rfloor = n$?

Let $(a_n)_{n\ge 1}$ be the sequence defined as the following : $$a_1=1 ,\ a_{n+1}=\dfrac{a_n}{n} + \dfrac{n}{a_n} ,\ n\ge1$$ Show that for every $n\ge4,\ \lfloor a_n^2 \rfloor = n$. My approach to ...