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Omni-MATH / There is an integer n>1. There are n² stations on a slope of a mountain, all at different altitudes. Each of two cable car companies, A and B, operates k cable cars; each cable car…

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problem

There is an integer n>1n > 1. There are n2n^2 stations on a slope of a mountain, all at different altitudes. Each of two cable car companies, AA and BB, operates kk cable cars; each cable car provides a transfer from one of the stations to a higher one (with no intermediate stops). The kk cable cars of AA have kk different starting points and kk different finishing points, and a cable car which starts higher also finishes higher. The same conditions hold for BB. We say that two stations are linked by a company if one can start from the lower station and reach the higher one by using one or more cars of that company (no other movements between stations are allowed). Determine the smallest positive integer kk for which one can guarantee that there are two stations that are linked by both companies. [i]
Plain-text mathematical notation (without MathML)
There is an integer n>1. There are n² stations on a slope of a mountain, all at different altitudes. Each of two cable car companies, A and B, operates k cable cars; each cable car provides a transfer from one of the stations to a higher one (with no intermediate stops). The k cable cars of A have k different starting points and k different finishing points, and a cable car which starts higher also finishes higher. The same conditions hold for B. We say that two stations are linked by a company if one can start from the lower station and reach the higher one by using one or more cars of that company (no other movements between stations are allowed). Determine the smallest positive integer k for which one can guarantee that there are two stations that are linked by both companies.

[i]
Original LaTeX notation
There is an integer $n > 1$. There are $n^2$ stations on a slope of a mountain, all at different altitudes. Each of two cable car companies, $A$ and $B$, operates $k$ cable cars; each cable car provides a transfer from one of the stations to a higher one (with no intermediate stops). The $k$ cable cars of $A$ have $k$ different starting points and $k$ different finishing points, and a cable car which starts higher also finishes higher. The same conditions hold for $B$. We say that two stations are linked by a company if one can start from the lower station and reach the higher one by using one or more cars of that company (no other movements between stations are allowed). Determine the smallest positive integer $k$ for which one can guarantee that there are two stations that are linked by both companies.

[i]

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Source and history

task_id
d6812b41-19cf-5141-9dd7-c55cf25f46c6
task_key
test--d6812b41-19cf-5141-9dd7-c55cf25f46c6
task_revision_id
1
config / split
/ test

Official source

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