Move out the floor

Algebra Level 4

x 2 x x \large\lfloor x^2\rfloor\space {\huge\square} \space x\lfloor x\rfloor What should we fill in if x > 0 x>0 ?

= = \le None of the others. \ge

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1 solution

Chew-Seong Cheong
Jun 27, 2018

If x x is an integer, then x 2 = x 2 \lfloor x^2 \rfloor = x^2 and x x = x 2 x\lfloor x\rfloor = x^2 x 2 = x x \implies \lfloor x^2 \rfloor = x\lfloor x\rfloor . If x = 1.3 x=1.3 , then 1. 3 2 = 1 \lfloor 1.3^2 \rfloor = 1 and 1.3 1.3 = 1.3 1.3 \lfloor 1.3 \rfloor = 1.3 x 2 < x x \implies \lfloor x^2 \rfloor < x\lfloor x\rfloor . If x = 1.5 x=1.5 , then 1. 5 2 = 2 \lfloor 1.5^2 \rfloor = 2 and 1.5 1.5 = 1.5 1.5 \lfloor 1.5 \rfloor = 1.5 x 2 > x x \implies \lfloor x^2 \rfloor > x\lfloor x\rfloor .

Therefore the answer is "None of the other."

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