1-2-x-1-2-2-x-2-3-2-x-3-100-2-x-100-10100-for-x-non-negative-integers-find-the-possible-value-of-x-

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

⌊ \frac{1}{2} x - 1 ⌋ + ⌊ \frac{2}{2} x - 2 ⌋ + ⌊ \frac{3}{2} x - 3 ⌋ + \dots + ⌊ \frac{100}{2} x - 100 ⌋ ⩽ 10100

for x non negative integers .

find the possible value of x

Answered by golsendro last updated on 05/Nov/24

\underset{―}{}

Commented by hardmath last updated on 05/Nov/24

thankyou dearprofessor

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

w h e n A ⩽ B i s g i v e n a n d y o u g e t

A ⩽ C, h o w c a n y o u t h e n s a y C ⩽ B,

n o t B ⩽ C ?

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

h o w c a n y o u g o t h e s t e p

f r o m ∣ z_{1} + z_{2} ∣⩽∣ z_{1} ∣ + ∣ z_{2} ∣

t o ⌊ z_{1} - z_{2} ⌋ ⩽ ⌊ z_{1} ⌋ - ⌊ z_{2} ⌋ ?

Commented by golsendro last updated on 06/Nov/24

what the correct answer ?

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

y o u r a n s w e r i s b y a c c i d e n t c o r r e c t,

b u t y o u r p a t h i s w r o n g .

Commented by golsendro last updated on 06/Nov/24

⌊ x - y ⌋ ⩾ ⌊ x ⌋ - ⌊ y ⌋

{it}^{'} s correct ?

⌊ \frac{1}{2} - 1 ⌋ = ⌊ - \frac{1}{2} ⌋ = - 1

⌊ \frac{1}{2} ⌋ - ⌊ 1 ⌋ = 0 - 1 = - 1

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

c a s e 1 : x = e v e n = 2 n w i t h n ⩾ 0

⌊ n - 1 ⌋ + ⌊ 2 n - 2 ⌋ + \dots + ⌊ 100 n - 100] ⩽ 10100

(1 + 2 + \dots + 100) (n - 1) ⩽ 10100

5050 (n - 1) ⩽ 10100

\Rightarrow n ⩽ \frac{10100}{5050} + 1 = 3

\Rightarrow x = 0, 2, 4, 6

c a s e 2 : x = o d d = 2 n + 1 w i t h n ⩾ 0

⌊ n + \frac{1}{2} - 1 ⌋ + ⌊ 2 n + 1 - 2 ⌋ + ⌊ 3 n + \frac{3}{2} - 3 ⌋ + \dots + ⌊ 100 n + 50 - 100] ⩽ 10100

(1 + 2 + \dots + 100) (n - 1) + ⌊ \frac{1}{2} ⌋ + ⌊ 1 ⌋ + ⌊ \frac{3}{2} ⌋ + \dots + ⌊ \frac{99}{2} ⌋ + ⌊ 50] ⩽ 10100

5050 (n - 1) + 0 + 2 \times 1 + 2 \times 2 + \dots + 2 \times 49 + 50 ⩽ 10100

5050 (n - 1) + 0 + 2 (1 + 2 + \dots + 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

s u m m a r y :

x = 0, 1, 2, 3, 4, 5, 6

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