Algebra

Floor and Ceiling Functions

Floor and Ceiling Functions: Level 4 Challenges

         

k=1202k = ? \large \sum_{k=1}^{202} \left \lceil \sqrt k \ \right \rceil = \ ?

Find the positive integer nn, for which log21+log22+log23++log2n=1994.\lfloor \log_2{1}\rfloor+\lfloor\log_2{2}\rfloor+\lfloor\log_2{3}\rfloor+\cdots+\lfloor\log_2{n}\rfloor=1994.

What is the minimum integer value of xx that satisfies the equation

xx+34=0?\lfloor{\sqrt{x}}\rfloor - \lfloor{\sqrt{x+34}}\rfloor=0 ?

This problem is posed by Siam H.

Details and assumptions

The function x:RZ\lfloor x \rfloor: \mathbb{R} \rightarrow \mathbb{Z} refers to the greatest integer smaller than or equal to xx. For example 2.3=2\lfloor 2.3 \rfloor = 2 and 5=5\lfloor -5 \rfloor = -5.

x+0.19+x+0.20+x+0.21++x+0.91=542\left\lfloor x+0.19 \right\rfloor +\left\lfloor x+0.20 \right\rfloor +\left\lfloor x+0.21 \right\rfloor + \ldots + \left\lfloor x+0.91 \right\rfloor =542

If xx satisfies the equation above, find the value of 100x\left\lfloor 100x \right\rfloor .

Note that X\left\lfloor X \right\rfloor denote the floor function of XX.

The number of real solutions of 7x+23{x}=1917\lfloor x\rfloor + 23\{x\}=191 is?

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