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Omni-MATH / Let S₀=0 and let S_(k) equal a₁+2a₂+…+ka_(k) for k≥1. Define a_(i) to be 1 if S_(i−1)<i and -1 if S_(i−1)≥i. What is the largest k≤2010 such that S_(k)=0?

Problem

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problem

Let S0=0S_{0}=0 and let SkS_{k} equal a1+2a2+…+kaka_{1}+2 a_{2}+\ldots+k a_{k} for k≥1k \geq 1. Define aia_{i} to be 1 if Si−1<iS_{i-1}<i and -1 if Si−1≥iS_{i-1} \geq i. What is the largest k≤2010k \leq 2010 such that Sk=0S_{k}=0?
Plain-text mathematical notation (without MathML)
Let S₀=0 and let S_(k) equal a₁+2a₂+…+ka_(k) for k≥1. Define a_(i) to be 1 if S_(i−1)<i and -1 if S_(i−1)≥i. What is the largest k≤2010 such that S_(k)=0?
Original LaTeX notation
Let $S_{0}=0$ and let $S_{k}$ equal $a_{1}+2 a_{2}+\ldots+k a_{k}$ for $k \geq 1$. Define $a_{i}$ to be 1 if $S_{i-1}<i$ and -1 if $S_{i-1} \geq i$. What is the largest $k \leq 2010$ such that $S_{k}=0$?

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Source and history

task_id
dd676528-949f-5490-ba7c-4658b6431168
task_key
test--dd676528-949f-5490-ba7c-4658b6431168
task_revision_id
1
config / split
/ test

Official source

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