# 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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### Find $x_{2017}$ where $x^{2}-\lfloor x^2 \rfloor = \left(x-\lfloor x \rfloor\right)^{2}$? [closed]

Let $x_{1}<x_{2}< x_{3}<\cdots$ be all real numbers greater than or equal to $1$ satisfying: $x^{2}-\lfloor x^2 \rfloor = \left(x-\lfloor x \rfloor\right)^{2}$. Then $x_{2017}$ can be ...
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### Solution-verification: Solve $3x^2-6x+4 = 6\{x\}\bigl(\lfloor x\rfloor - \{x\}\bigr)$

the problem Solve in the set of real numbers the following equation $$3x^2-6x+4 = 6\{x\}\bigl(\lfloor x\rfloor - \{x\}\bigr),$$ where $\lfloor x\rfloor$ and $\{x\}$ are the whole part and the ...
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1 vote
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### Prove/disprove $\sum_{i=1}^n \left\lfloor {x_i}^{y} \right\rfloor \leq \left\lfloor x^y\right\rfloor$ for $x_i,y\geq 1.$

If $x_i,y\in\mathbb{R}_{\geq 1},\ \displaystyle\sum_{i=1}^n x_i = x,$ then it is obviously true that $\displaystyle \sum_{i=1}^n {x_i}^y \leq x^y,$ due to the Binomial theorem. After trying various ...
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### Solving $a = x \mod b$ without congruences [duplicate]

I have never dealt with congruences, and I want to stay as pragmatic as possible here. Using a similar approach to trigonometric equations, where the solution is rather a set of answers that fulfil ...
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### What is my reasoning error when solving this compound inequality using floor bounds?

I'm looking to solve the compound inequality $$(2/3)^hn \lt n_0 \le (2/3)^{h-1}n$$ for the integer h, with $0 \lt n_0 \le n$. Context $h$ represents the height of the recursion-tree of a divide-and-...
1 vote
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1 vote
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### Simplify $\lfloor \frac{k}{10}\rfloor\cdot\lfloor \frac{k-10\cdot\lfloor \frac{k}{10}\rfloor}{5}\rfloor$

Recently when learning SICP, I reread exercises done before (about 20) after schemewiki is resumed with some updates. I read this code which is the reference of sicp-solutions which is the reference ...
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10 views

### question regarding modelling motion of particle using a step function

I have a particle which starts off with a velocity of 30 meter per second and reduces its velocity by 5 meter per second instantly every two seconds i thought a good way to model this kind of motion ...
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### Proving $a_{n} = \lfloor \sqrt{2n} + \frac{1}{2} \rfloor$ for sequence $1, 2, 2, 3, 3, 3, ...$ where positive integer $k$ is repeated $k$ times.

Rosen [ Discrete Math ] Exercise 28 : Proving $a_{n} = \lfloor \sqrt{2n} + \frac{1}{2} \rfloor$ for sequence $1, 2, 2, 3, 3, 3, ...$ where positive integer $k$ is repeated $k$ times. A Solution is ...
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