# 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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### Limit of an Integral with a Floor Function

I'm looking to evaluate the following integral and would like to verify my answer: $$\lim_{n\to\infty} \frac{1}{n^2} \int_{0}^{n^2} \sqrt{\frac{n^2-\lfloor \sqrt{x}\rfloor^2}{x}} \ \textrm{d}x$$ I've ...
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### Confused on answer given for the question: prove $\lceil\frac{n}{m}\rceil = \lfloor\frac{n+m-1}{m}\rfloor$

1https://math.stackexchange.com/a/1281757/801877 Surprisingly, when trying to prove this problem I tried doing this exact way and got piecewise definition for ceiling(n/m), but I'm not sure how to do ...
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### Floor function and Golden ratio

Consider implicit equation : $$x=\left(\left(y^{-⌊x⌋}+x\right)^{-\frac{1}{⌊x⌋}}-y\right)\left(\sqrt{y^{-⌊x⌋}+x}+y\right)$$ I have observed that $$\text{when} \ x \to 0^{-}, \ y \to \frac{1}{\Phi^{2}}$$...
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### $\left[\dfrac 12\displaystyle\sum_{n=1}^{k^2}\frac 1{\sqrt n}\right]=k-1$

Consider this $\left[\dfrac 12\displaystyle\sum_{n=1}^{k^2}\frac 1{\sqrt n}\right]$ where $[\cdot]$ is the greatest integer function. I had observed that its value is $(k-1)$ by putting $k=2,3,4,$ ...
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### Accumulation points of the sequence $c_n = \lfloor \cos(\sqrt{n}) \rfloor$

I'm trying to find the accumulation points for the sequence $c_n = \lfloor \cos(\sqrt{n}) \rfloor, n \in \mathbb{N}_0$. I know what the points are, but I'm having trouble coming up with an explicit ...
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### What is $\int_{0}^{\pi/2}\frac{\operatorname{lcm}(a\cos x,a\sin x)}{a^2}dx$?

I came up with this while messing around with the $\gcd$ and $\operatorname{lcm}$ functions in Desmos. $$I(a)=\int_{0}^{\pi/2}\frac{\operatorname{lcm}(a\cos x,a\sin x)}{a^2}dx$$ The function inside ...
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### Challenge: Define rounding functions using integer checking functions

This question was inspired by Willans' formula, in which he used the cosine function as a way to constrain a number and floor it to be able to count primes (see https://www.youtube.com/watch?v=...
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### Finding the sum of the floor function of $a,(b-1)/2,c$ given two symmetric sums

Problem: Let $a<b<c$ be $3$ real numbers satisfying $a+b+c=6$, $ab+bc+ca=9$. Then, determine the value of $\lfloor{a}\rfloor+\lfloor\frac{b-1}{2}\rfloor+\lfloor{c}\rfloor$. My method of solution:...
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### Calculate the whole part of $A=\frac{1012^2}{\sqrt{1*2}+\sqrt{3*4}+\cdots+\sqrt{2023*2024}}$

the question Calculate the whole part of: $$A=\frac{1012^2}{\sqrt{1*2}+\sqrt{3*4}+\cdots+\sqrt{2023*2024}}$$ my idea My first thought was to to fit the number between 2 consecutive natural numbers ...
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### Proving the Greatest Integer Fucntion (G.I.F) of (50!/e + 1/2) is D50(derangement)

Prove that, If D(n) denotes dearrangement for n objects, then D(50) = [ 50/e! + 1/2 ] Where [.] denotes the greatest integer function. Through exapnsion i generated 50/e! but how can we adjust the &...
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### Points of discontinuity of greatest Integer function

So here's my problem Find the points of discontinuity of $[x^2]$ in the interval$[-1,2]$ where [] is the G.I.F. I am not able to understand why at $2^{1/2}$ and $3^{1/2}$ the function is being ...
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### Quesiton regarding nature of equations involving fractional part and greatest integer functions

This question is in context of the following problem Solve: $[x]^2 = x + 2\{x\}$ Where $[.]$ and $\{.\}$ represent the greatest integer and fractional part function respectively. The solution for the ...
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### Question regarding intersection of values of a function

This question is being asked in regards to the following problem Solve: $x^2 - 4 - \lfloor x \rfloor = 0$ The solution for the above equation in my textbook is done by drawing the graph of both the ...
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### Prove⌈a/b⌉ ≤ a/b + (b-1)/b [closed]

For integers $a, b > 0$, Prove $⌈a/b⌉ ≤ (a + (b-1))/b$ RHS $= a/b + (b-1)/b$ where $(b-1)/b$ is $[0,1)$ If $a/b$ is an integer, inequality holds true as we are adding non-negative term. If $a/b$ ...
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### For $0<x<1$, let $f(x)=\int_0^1\left( \left\lfloor\frac{x}{y}\right\rfloor-x \left\lfloor\frac{1}{y}\right\rfloor\right)dy\ldots$

For $0<x<1$, let $f(x)=\int_0^1\left( \left\lfloor\frac{x}{y}\right\rfloor-x \left\lfloor\frac{1}{y}\right\rfloor\right)dy$, $\lfloor. \rfloor$ denotes the greatest integer function. If $f(x)$ ...
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### Solve in $\mathbb{R}^+$ algebra calculus Equation with derivatives

I was trying to solve this question: Solve in $\mathbb{R}^+$ the Equation $7^x-5^x=\lfloor x^2 \rfloor+1$ Where $\lfloor t\rfloor$ is the floor function It is known that $\lfloor x^2\rfloor\leq x^2$ ...
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### Prove that the maximum difference between $\frac{n}{6}$ and $\lfloor \frac{(n-3)}{6} \rfloor$ is $1 \frac{1}{3}$?

Prove that the maximum difference between $\frac{n}{6}$ and $\lfloor \frac{(n-3)}{6} \rfloor$ where $n$ is an integer is $1 \frac{1}{3}$? I know this is true because for $n = 8 + 6i$ where $i$ is an ...
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### Is there a nice form to $\sum_{k=1}^{n}\left\lfloor\sqrt{ r^2-k^2 }\right\rfloor$?

$$\sum_{k=1}^{n}\left\lfloor\sqrt{ r^2-k^2 }\right\rfloor$$ where $r$ is a constant (not necessarily an integer). Note that $r\ge n$ and both $r, n$ are positive. I apologize if I'm not able to ...
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### Integer part of $x^n$

Given a real number $x>1$ and a natural number $n$, what can we say about the integer part of $x^n$ in terms of $x$ and $n$? For simplicity, let us assume $x<2$. For the first few values of $n$, ...
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### Is there a spelling mistake or am I missing something

Here, $[ \cdot]$ is $\lfloor \cdot \rfloor$ floor function. N $\in N$. Where did $\frac{[Nx]} N + \frac{1}{2N}$ came from and how does $x$ differs by $\frac{1}{2N}$. Shouldn't it be $\frac{1}N$ if ...
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### Find the minimum value of $(x-y)^2+(x-y+ \frac{1}{y}-\frac{1}{x})^2$ where $x>0>y$

If $\lambda$ denotes the minimum value of $(x-y)^2+(x-y+ \frac{1}{y}-\frac{1}{x})^2$ where $x>0>y$, then find the value of [3$\lambda$] where [.] denotes the greatest integer function. My ...
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