Floor Function
Read this topicExplanations 4
- Floor and ceiling functions Say we have α\alphaα, then ⌊α⌋\lfloor\alpha\rfloor⌊α⌋ is the greatest integer less than or equal to α\alphaα and ⌈α⌉\lceil\alpha\rceil⌈α⌉ is the smal…
By Hyunwoo (David) Lee, Ahaan Rungta, Expii Team
- Floor Function Floor function of xxx where xxx is a real number, denoted by ⌊x⌋\lfloor x\rfloor⌊x⌋ is the largest integer less than or equal to xxx. For example, ⌊3.1415⌋=3\lflo…
By amaterasu, Alex Federspiel
- Prove that for real xxx and integer nnn, we have ⌊x+n⌋=⌊x⌋+n\lfloor x+n\rfloor=\lfloor x\rfloor+n⌊x+n⌋=⌊x⌋+n.
By amaterasu
- Prove that for real x,yx,yx,y, we have ⌊x+y⌋≥⌊x⌋+⌊y⌋\lfloor x+y\rfloor\geq\lfloor x\rfloor+\lfloor y\rfloor⌊x+y⌋≥⌊x⌋+⌊y⌋.
By amaterasu
Problems 3
- Determine the absolute value of the sum ⌊2013sin0∘⌋+⌊2013sin1∘⌋+⋯+⌊2013sin359∘⌋,\lfloor 2013\sin{0^\circ} \rfloor + \lfloor 2013\sin{1^\circ} \rfloor + \cdots + \lfloor 2013\…
- Let f(x)=∑k=210(⌊kx⌋−k⌊x⌋)f(x)=\sum_{k=2}^{10}(\lfloor kx \rfloor -k \lfloor x \rfloor)f(x)=∑k=210(⌊kx⌋−k⌊x⌋), where ⌊r⌋\lfloor r \rfloor⌊r⌋ denotes the greatest integer less t…
- ⌊x99⌋=⌊x101⌋\left \lfloor \dfrac{x}{99}\right \rfloor = \left \lfloor \dfrac{x}{101}\right \rfloor⌊99x⌋=⌊101x⌋ Find the number of positive integers xxx with satisfy the above …