Question and Answers Forum

All Questions      Topic List

Integration Questions

Previous in All Question      Next in All Question      

Previous in Integration      Next in Integration      

Question Number 213423 by efronzo1 last updated on 05/Nov/24

$$\lfloor \frac{1}{2}\mathrm{x}-1\rfloor +\lfloor \frac{2}{2}\mathrm{x}-2\rfloor +\lfloor \frac{3}{2}\mathrm{x}-3\rfloor +...+\lfloor \frac{100}{2}\mathrm{x}-100\rfloor ⩽10100$$$$\mathrm{for}\mathrm{x}\mathrm{non}\mathrm{negative}\mathrm{integers}.$$$$\mathrm{find}\mathrm{the}\mathrm{possible}\mathrm{value}\mathrm{of}\mathrm{x}$$

Answered by golsendro last updated on 05/Nov/24

$$\underset{―}{}$$

Commented by hardmath last updated on 05/Nov/24

$$\mathrm{thankyou}\mathrm{dearprofessor}$$

Commented by mr W last updated on 05/Nov/24

$$whenA⩽Bisgivenandyouget$$$$A⩽C,howcanyouthensayC⩽B,$$$$notB⩽C?$$

Commented by mr W last updated on 06/Nov/24

$$howcanyougothestep$$$$from\mid z_1+z_2\mid ⩽\mid z_1\mid +\mid z_2\mid$$$$to\lfloor z_1-z_2\rfloor ⩽\lfloor z_1\rfloor -\lfloor z_2\rfloor ?$$

Commented by golsendro last updated on 06/Nov/24

$$\mathrm{what}\mathrm{the}\mathrm{correct}\mathrm{answer}?$$

Commented by mr W last updated on 06/Nov/24

$$youranswerisbyaccidentcorrect,$$$$butyourpathiswrong.$$

Commented by golsendro last updated on 06/Nov/24

$$\lfloor \mathrm{x}-\mathrm{y}\rfloor ⩾\lfloor \mathrm{x}\rfloor -\lfloor \mathrm{y}\rfloor$$$${\mathrm{it}}^{\prime }\mathrm{s}\mathrm{correct}?$$$$\lfloor \frac{1}{2}-1\rfloor =\lfloor -\frac{1}{2}\rfloor =-1$$$$\lfloor \frac{1}{2}\rfloor -\lfloor 1\rfloor =0-1=-1$$

Answered by mr W last updated on 06/Nov/24

$$\mathit{c}\mathit{a}\mathit{s}\mathit{e}1:x=even=2nwithn⩾0$$$$\lfloor n-1\rfloor +\lfloor 2n-2\rfloor +...+\lfloor 100n-100]⩽10100$$$$(1+2+...+100)(n-1)⩽10100$$$$5050(n-1)⩽10100$$$$\Rightarrow n⩽\frac{10100}{5050}+1=3$$$$\Rightarrow x=0,2,4,6$$$$\mathit{c}\mathit{a}\mathit{s}\mathit{e}2:x=odd=2n+1withn⩾0$$$$\lfloor n+\frac{1}{2}-1\rfloor +\lfloor 2n+1-2\rfloor +\lfloor 3n+\frac{3}{2}-3\rfloor +...+\lfloor 100n+50-100]⩽10100$$$$(1+2+...+100)(n-1)+\lfloor \frac{1}{2}\rfloor +\lfloor 1\rfloor +\lfloor \frac{3}{2}\rfloor +...+\lfloor \frac{99}{2}\rfloor +\lfloor 50]⩽10100$$$$5050(n-1)+0+2\times 1+2\times 2+...+2\times 49+50⩽10100$$$$5050(n-1)+0+2(1+2+...+49+50)-50⩽10100$$$$5050(n-1)+50\times 51-50⩽10100$$$$5050(n-1)+2500⩽10100$$$$\Rightarrow n⩽\frac{10100-2500}{5050}+1\approx 2.505\Rightarrow n=0,1,2$$$$\Rightarrow x=1,3,5$$$$summary:$$$$x=0,1,2,3,4,5,6$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com