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Omni-MATH / In the game of Galactic Dominion, players compete to amass cards, each of which…

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problem

In the game of Galactic Dominion, players compete to amass cards, each of which is worth a certain number of points. Say you are playing a version of this game with only two kinds of cards, planet cards and hegemon cards. Each planet card is worth 2010 points, and each hegemon card is worth four points per planet card held. You start with no planet cards and no hegemon cards, and, on each turn, starting at turn one, you take either a planet card or a hegemon card, whichever is worth more points given the hand you currently hold. Define a sequence {an}\left\{a_{n}\right\} for all positive integers nn by setting ana_{n} to be 0 if on turn nn you take a planet card and 1 if you take a hegemon card. What is the smallest value of NN such that the sequence aN,aN+1,…a_{N}, a_{N+1}, \ldots is necessarily periodic (meaning that there is a positive integer kk such that an+k=ana_{n+k}=a_{n} for all n≥N)\left.n \geq N\right) ?
Plain-text mathematical notation (without MathML)
In the game of Galactic Dominion, players compete to amass cards, each of which is worth a certain number of points. Say you are playing a version of this game with only two kinds of cards, planet cards and hegemon cards. Each planet card is worth 2010 points, and each hegemon card is worth four points per planet card held. You start with no planet cards and no hegemon cards, and, on each turn, starting at turn one, you take either a planet card or a hegemon card, whichever is worth more points given the hand you currently hold. Define a sequence {a_(n)} for all positive integers n by setting a_(n) to be 0 if on turn n you take a planet card and 1 if you take a hegemon card. What is the smallest value of N such that the sequence a_(N),a_(N+1),… is necessarily periodic (meaning that there is a positive integer k such that a_(n+k)=a_(n) for all n≥N) ?
Original LaTeX notation
In the game of Galactic Dominion, players compete to amass cards, each of which is worth a certain number of points. Say you are playing a version of this game with only two kinds of cards, planet cards and hegemon cards. Each planet card is worth 2010 points, and each hegemon card is worth four points per planet card held. You start with no planet cards and no hegemon cards, and, on each turn, starting at turn one, you take either a planet card or a hegemon card, whichever is worth more points given the hand you currently hold. Define a sequence $\left\{a_{n}\right\}$ for all positive integers $n$ by setting $a_{n}$ to be 0 if on turn $n$ you take a planet card and 1 if you take a hegemon card. What is the smallest value of $N$ such that the sequence $a_{N}, a_{N+1}, \ldots$ is necessarily periodic (meaning that there is a positive integer $k$ such that $a_{n+k}=a_{n}$ for all $\left.n \geq N\right)$ ?

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task_id
a3e02f33-3dfc-5ece-b4ad-866bf32cd5e1
task_key
test--a3e02f33-3dfc-5ece-b4ad-866bf32cd5e1
task_revision_id
1
config / split
/ test

Official source

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