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Omni-MATH / For each real number x, let

Problem

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problem

For each real number xx, let f(x)=∑n∈Sx12n, f(x) = \sum_{n\in S_x} \frac{1}{2^n}, where SxS_x is the set of positive integers nn for which ⌊nx⌋\lfloor nx \rfloor is even. What is the largest real number LL such that f(x)≥Lf(x) \geq L for all x∈[0,1)x \in [0,1)? (As usual, ⌊z⌋\lfloor z \rfloor denotes the greatest integer less than or equal to zz.)
Plain-text mathematical notation (without MathML)
For each real number x, let
f(x)=∑_(n∈S_(x))(1)/(2^(n)),
where S_(x) is the set of positive integers n for which ⌊nx⌋ is even. What is the largest real number L such that f(x)≥L for all x∈[0,1)? (As usual, ⌊z⌋ denotes the greatest integer less than or equal to z.)
Original LaTeX notation
For each real number $x$, let
\[
f(x) = \sum_{n\in S_x} \frac{1}{2^n},
\]
where $S_x$ is the set of positive integers $n$ for which $\lfloor nx \rfloor$ is even. What is the largest real number $L$ such that $f(x) \geq L$ for all $x \in [0,1)$? (As usual, $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$.)

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Source and history

task_id
b98bfe01-f483-52b2-a4ff-acd5a91263b0
task_key
test--b98bfe01-f483-52b2-a4ff-acd5a91263b0
task_revision_id
1
config / split
/ test

Official source

initial import