Floor Ceiling Ceiling Floor: JEE Style

Algebra Level 5

xx+xx=1729 \large \lceil x \lfloor x \rfloor \rceil + \lfloor x \lceil x \rceil \rfloor = 1729

If the range of positive xx satisfy the equation above is αxβ \alpha \leq x \leq \beta .

What is the value of 30α+29β+130\alpha + 29 \beta+1 if we know that it is an integer?

Inspired by: Floor Ceiling Ceiling Floor.

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