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Omni-MATH / Turbo the snail sits on a point on a circle with circumference 1. Given an infinite sequence of positive real numbers c₁,c₂,c₃,…, Turbo successively crawls distances c₁,c₂,c₃,… aro…

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problem

Turbo the snail sits on a point on a circle with circumference 11. Given an infinite sequence of positive real numbers c1,c2,c3,…c_1, c_2, c_3, \dots, Turbo successively crawls distances c1,c2,c3,…c_1, c_2, c_3, \dots around the circle, each time choosing to crawl either clockwise or counterclockwise. Determine the largest constant C>0C > 0 with the following property: for every sequence of positive real numbers c1,c2,c3,…c_1, c_2, c_3, \dots with ci<Cc_i < C for all ii, Turbo can (after studying the sequence) ensure that there is some point on the circle that it will never visit or crawl across.
Plain-text mathematical notation (without MathML)
Turbo the snail sits on a point on a circle with circumference 1. Given an infinite sequence of positive real numbers c₁,c₂,c₃,…, Turbo successively crawls distances c₁,c₂,c₃,… around the circle, each time choosing to crawl either clockwise or counterclockwise.
Determine the largest constant C>0 with the following property: for every sequence of positive real numbers c₁,c₂,c₃,… with c_(i)<C for all i, Turbo can (after studying the sequence) ensure that there is some point on the circle that it will never visit or crawl across.
Original LaTeX notation
Turbo the snail sits on a point on a circle with circumference $1$. Given an infinite sequence of positive real numbers $c_1, c_2, c_3, \dots$, Turbo successively crawls distances $c_1, c_2, c_3, \dots$ around the circle, each time choosing to crawl either clockwise or counterclockwise.
Determine the largest constant $C > 0$ with the following property: for every sequence of positive real numbers $c_1, c_2, c_3, \dots$ with $c_i < C$ for all $i$, Turbo can (after studying the sequence) ensure that there is some point on the circle that it will never visit or crawl across.

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Source and history

task_id
f4b6d1d9-93e5-526e-a6d4-20f76f33042c
task_key
test--f4b6d1d9-93e5-526e-a6d4-20f76f33042c
task_revision_id
1
config / split
/ test

Official source

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