MathDB

2017 A5: Solution Set to Floor Equation

Source:

January 29, 2017

2017algebra

Problem Statement

The set

S

of positive real numbers

x

such that

\left\lfloor\frac{2x}{5}\right\rfloor + \left\lfloor\frac{3x}{5}\right\rfloor + 1 = \left\lfloor x\right\rfloor

can be written as

S = \bigcup_{j = 1}^{\infty} I_{j}

, where the

I_{i}

are disjoint intervals of the form

[a_{i}, b_{i}) = \{x \, | \, a_i \leq x < b_i\}

and

b_{i} \leq a_{i+1}

for all

i \geq 1

. Find

\sum_{i=1}^{2017} (b_{i} - a_{i})

.

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