Cute challenge problem!

Question

kainui · Answer

What are all the real numbers x that satisfy this equation: $$\lfloor x \rfloor^5 + \{x \}^5 = x^5$$

The function $f(x)=\lfloor x\rfloor$ rounds down to the nearest whole number and the function $g(x) = \{x\}$ gives only the fractional part of a number, so quick examples:

$$\lfloor 3.2 \rfloor = 2$$$$\{9.732 \} = .732$$

Good luck! :D

mathstudent55 · Answer

$\lfloor 3.2 \rfloor = 2$   ???

kainui · Answer

uhhh typo hahaha should equal 3

sparrow2 · Answer

for [-2,3]=-2?

sparrow2 · Answer

i think it works for only integers and numbers for [-1,1]

mathstudent55 · Answer

Every integer is a solution.

sparrow2 · Answer

alos of[-1,1] because rounnd part is 0

mathstudent55 · Answer

$x = -0.5$

$\lfloor x \rfloor^5 + \{x \}^5 = x^5$

$\lfloor -0.5 \rfloor^5 + \{-0.5 \}^5 = (-0.5)^5$

$(-1)^5 + (0.5)^5 = (-0.5)^5$

$-1 + (0.5)^5 \ne (-0.5)^5$

Perhaps it's all integers and [0,1].

sparrow2 · Answer

yeah, it depends how you round the negative numbers.

mathstudent55 · Answer

We don't have a choice.
"The function f(x)=⌊x⌋ rounds down to the nearest whole number and the "

kainui · Answer

You're on the right track, all integers and the interval [0,1] are definitely special cases for considering in this problem :)

thomas5267 · Answer

What would $\{-1.5\}$ be?  I was thinking something along the lines of $\lfloor x\rfloor^5+\{x\}^5=\left(\lfloor x\rfloor+\{x\}\right)^5$

kainui · Answer

@thomas5267  
I will just go ahead and confirm it by saying this statement is true:

$$x = \lfloor x \rfloor + \{x\}$$

candi5a18 · Answer

ur cute

candi5a18 · Answer

:D

candi5a18 · Answer

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