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Omni-MATH / Let k be a positive integer. Scrooge McDuck owns k gold coins. He also owns infinitely many boxes B₁,B₂,B₃,… Initially, bow B₁ contains one coin, and the k−1 other coins are on McD…

Problem

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problem

Let kk be a positive integer. Scrooge McDuck owns kk gold coins. He also owns infinitely many boxes B1,B2,B3,…B_1, B_2, B_3, \ldots Initially, bow B1B_1 contains one coin, and the k−1k-1 other coins are on McDuck's table, outside of every box. Then, Scrooge McDuck allows himself to do the following kind of operations, as many times as he likes: - if two consecutive boxes BiB_i and Bi+1B_{i+1} both contain a coin, McDuck can remove the coin contained in box Bi+1B_{i+1} and put it on his table; - if a box BiB_i contains a coin, the box Bi+1B_{i+1} is empty, and McDuck still has at least one coin on his table, he can take such a coin and put it in box Bi+1B_{i+1}. As a function of kk, which are the integers nn for which Scrooge McDuck can put a coin in box BnB_n?
Plain-text mathematical notation (without MathML)
Let k be a positive integer. Scrooge McDuck owns k gold coins. He also owns infinitely many boxes B₁,B₂,B₃,… Initially, bow B₁ contains one coin, and the k−1 other coins are on McDuck's table, outside of every box.
Then, Scrooge McDuck allows himself to do the following kind of operations, as many times as he likes:
- if two consecutive boxes B_(i) and B_(i+1) both contain a coin, McDuck can remove the coin contained in box B_(i+1) and put it on his table;
- if a box B_(i) contains a coin, the box B_(i+1) is empty, and McDuck still has at least one coin on his table, he can take such a coin and put it in box B_(i+1).
As a function of k, which are the integers n for which Scrooge McDuck can put a coin in box B_(n)?
Original LaTeX notation
Let $k$ be a positive integer. Scrooge McDuck owns $k$ gold coins. He also owns infinitely many boxes $B_1, B_2, B_3, \ldots$ Initially, bow $B_1$ contains one coin, and the $k-1$ other coins are on McDuck's table, outside of every box.
Then, Scrooge McDuck allows himself to do the following kind of operations, as many times as he likes:
- if two consecutive boxes $B_i$ and $B_{i+1}$ both contain a coin, McDuck can remove the coin contained in box $B_{i+1}$ and put it on his table;
- if a box $B_i$ contains a coin, the box $B_{i+1}$ is empty, and McDuck still has at least one coin on his table, he can take such a coin and put it in box $B_{i+1}$.
As a function of $k$, which are the integers $n$ for which Scrooge McDuck can put a coin in box $B_n$?

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Source and history

task_id
bafa85b8-88ae-5898-954a-a879280999f5
task_key
test--bafa85b8-88ae-5898-954a-a879280999f5
task_revision_id
1
config / split
/ test

Official source

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