find the smallest integer values of $x$ for which $\lfloor \sqrt {x +15} \rfloor - \lfloor \sqrt x \rfloor ═0$ holds

plz solve that or tell me where i can

## Floors and roots

### Re: Floors and roots

Hi ibnesina, Welcome to BdMO forum, please read and try to follow this guide http://www.matholympiad.org.bd/forum/vi ... p?f=25&t=2 on using LaTeX for writing mathematical symbols or equations.

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Use $L^AT_EX$, It makes our work a lot easier!

Nur Muhammad Shafiullah | Mahi

### Re: Floors and roots

you can check one by one. but the another way is that to check only squire number. Now notethat,

$x=4$ implies

$\lfloor \sqrt {x +15} \rfloor - \lfloor \sqrt x \rfloor ═1$

because $16<19<25$ so $ 4 <\sqrt { 4+15} <5$

now checking one by one squire number you will see that for $x=49$ you can calculate this without floor function

$ \sqrt {49 +15} - \sqrt 49 ═1$

then check $x=64$ you will get your answer.

actually according to experience the trick is that, just try to make a squire for both floor function and then the next number could be your answer. just check.

$x=4$ implies

$\lfloor \sqrt {x +15} \rfloor - \lfloor \sqrt x \rfloor ═1$

because $16<19<25$ so $ 4 <\sqrt { 4+15} <5$

now checking one by one squire number you will see that for $x=49$ you can calculate this without floor function

$ \sqrt {49 +15} - \sqrt 49 ═1$

then check $x=64$ you will get your answer.

actually according to experience the trick is that, just try to make a squire for both floor function and then the next number could be your answer. just check.

*হার জিত চিরদিন থাকবেই*

তবুও এগিয়ে যেতে হবে.........

বাধা-বিঘ্ন না পেরিয়ে

বড় হয়েছে কে কবে.........তবুও এগিয়ে যেতে হবে.........

বাধা-বিঘ্ন না পেরিয়ে

বড় হয়েছে কে কবে.........

### Re: Floors and roots

for $k\ge0$, $\lfloor\sqrt x\rfloor=\lfloor \sqrt{x+k}\rfloor$ if $k<2x+1$. So if $15<2x+1$ is satisfied by $8$ for smallest such $x$.ibnesina wrote:find the smallest integer values of $x$ for which $\lfloor \sqrt {x +15} \rfloor - \lfloor \sqrt x \rfloor ═0$ holds

plz solve that or tell me where i can

One one thing is neutral in the universe, that is $0$.

### Re: Floors and roots

You need to explain why.sm.joty wrote:but the another way is that to check only square number

One one thing is neutral in the universe, that is $0$.