x-2-x-2-100-x-R-min-x-

Question Number 187091 by Humble last updated on 13/Feb/23

⌊ x^{2} ⌋ - ⌊ x ⌋^{2} = 100

x \in R

m i n (x) = ?

Answered by MJS_new last updated on 13/Feb/23

x = n + f; n \in Z \land 0 ⩽ f < 1

⌊ x^{2} ⌋ - ⌊ x ⌋^{2} = 100

⌊ n^{2} + 2 n f + f^{2} ⌋ - ⌊ n + f ⌋^{2} = 100

n^{2} + ⌊ 2 n f + f^{2} ⌋ - n^{2} = 100

⌊ 2 n f + f^{2} ⌋ = 100

2 n f + f^{2} - 100 = 0 \land 0 ⩽ f < 1 \land n \in Z

\Rightarrow f = - n + \sqrt{n^{2} + 100}

0 ⩽ - n + \sqrt{n^{2} + 100} < 1

\Rightarrow n ⩾ 50

n = 50 \Rightarrow f = - 50 + 10 \sqrt{26}

x = n + f = 10 \sqrt{26}

Commented by Humble last updated on 13/Feb/23

t h a n k y o u, s i r

Commented by MJS_new last updated on 13/Feb/23

{you}^{'} re welcome

Answered by mr W last updated on 13/Feb/23

x = n + f

⌊ n^{2} + 2 n f + f^{2} ⌋ - ⌊ n + f ⌋^{2} = 100

n^{2} + ⌊ 2 n f + f^{2} ⌋ - n^{2} = 100

100 = ⌊ 2 n f + f^{2} ⌋ < 2 n + 1 \Rightarrow n > \frac{99}{2} \Rightarrow n ⩾ 50

100 ⩽ 2 n f + f^{2} < 101

f^{2} + 2 n f - 100 ⩾ 0

\Rightarrow f ⩾ - n + \sqrt{n^{2} + 100}

f^{2} + 2 n f - 101 < 0

\Rightarrow f < - n + \sqrt{n^{2} + 101}

g e n e r a l s o l u t i o n :

\sqrt{n^{2} + 100} ⩽ x < \sqrt{n^{2} + 101}

(n ⩾ 50)

m i n (x) = \sqrt{50^{2} + 100} = 10 \sqrt{26}

Commented by Humble last updated on 13/Feb/23

t h a n k y o u, B o s s .

Commented by mr W last updated on 13/Feb/23

y o u t o o!

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