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It is clear that the area must be a polygon with stakes at corners.

I do not have prove, but most probably:

  1. The fastest way to put stake it each corner is to follow perimeter of the polygon. So we must optimise an area between the path and a line (which goes through first and last stake). The maximum area is given by semicircle. (It is the shape of soap bubble put on a table, physical lows there optimise area of the bubble with given volume, which is virtually what we need). The aria would be $S = \pi(\frac{n}{\pi})^2 = n^2/\pi$.

  2. We should use regular polygon here with some number of corners $k$. But we must follow only half of it, similarly to semicircle (thanks kainekaine for insight on this). So side of polygon must be $A = (n-k/2\cdot m)/(k/2) = 2n/k-m$. The area of half polygon would be $S = 1/8(2n/k-m)^2cot(180/k)$. One needs to find derivative and assign it to zero to find the maximum. Unfortunately I can't do it without computer, but what have to be done is quite clear, may be some one else would like to do it.

  3. Here the strategy must depend strongly on probability distribution of the given time $t$, which is not given [Here I talk about initial formulation of the question, now it is changed].
    For standard uniform distribution from 0 to $T_{max}$, I don't know how to approach it strictly, but the wild guess would be to use logarithmic spiral, like in the task "What is quickest way out from forest, if you lost?".

It is clear that the area must be a polygon with stakes at corners.

I do not have prove, but most probably:

  1. The fastest way to put stake it each corner is to follow perimeter of the polygon. So we must optimise an area between the path and a line (which goes through first and last stake). The maximum area is given by semicircle. (It is the shape of soap bubble put on a table, physical lows there optimise area of the bubble with given volume, which is virtually what we need). The aria would be $S = \pi(\frac{n}{\pi})^2 = n^2/\pi$.

  2. We should use regular polygon here with some number of corners $k$. But we must follow only half of it, similarly to semicircle (thanks kaine for insight on this). So side of polygon must be $A = (n-k/2\cdot m)/(k/2) = 2n/k-m$. The area of half polygon would be $S = 1/8(2n/k-m)^2cot(180/k)$. One needs to find derivative and assign it to zero to find the maximum. Unfortunately I can't do it without computer, but what have to be done is quite clear, may be some one else would like to do it.

  3. Here the strategy must depend strongly on probability distribution of the given time $t$, which is not given [Here I talk about initial formulation of the question, now it is changed].
    For standard uniform distribution from 0 to $T_{max}$, I don't know how to approach it strictly, but the wild guess would be to use logarithmic spiral, like in the task "What is quickest way out from forest, if you lost?".

It is clear that the area must be a polygon with stakes at corners.

I do not have prove, but most probably:

  1. The fastest way to put stake it each corner is to follow perimeter of the polygon. So we must optimise an area between the path and a line (which goes through first and last stake). The maximum area is given by semicircle. (It is the shape of soap bubble put on a table, physical lows there optimise area of the bubble with given volume, which is virtually what we need). The aria would be $S = \pi(\frac{n}{\pi})^2 = n^2/\pi$.

  2. We should use regular polygon here with some number of corners $k$. But we must follow only half of it, similarly to semicircle (thanks kaine for insight on this). So side of polygon must be $A = (n-k/2\cdot m)/(k/2) = 2n/k-m$. The area of half polygon would be $S = 1/8(2n/k-m)^2cot(180/k)$. One needs to find derivative and assign it to zero to find the maximum. Unfortunately I can't do it without computer, but what have to be done is quite clear, may be some one else would like to do it.

  3. Here the strategy must depend strongly on probability distribution of the given time $t$, which is not given [Here I talk about initial formulation of the question, now it is changed].
    For standard uniform distribution from 0 to $T_{max}$, I don't know how to approach it strictly, but the wild guess would be to use logarithmic spiral, like in the task "What is quickest way out from forest, if you lost?".

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It is clear that the area must be a polygon with stakes at corners.

I do not have prove, but most probably:

  1. The fastest way to put stake it each corner is to follow perimeter of the polygon. So we must optimise an area between the path and a line (which goes through first and last stake). The maximum area is given by semicircle. (It is the shape of soap bubble put on a table, physical lows there optimise area of the bubble with given volume, which is virtually what we need). The aria would be $S = \pi(\frac{n}{\pi})^2 = n^2/\pi$.

  2. We should use regular polygon here with some number of corners $k$ and. But we must follow only half of it, similarly to semicircle (thanks kaine for insight on this). So side of polygon must be $A = (n-(k-1)\cdot m)/k$$A = (n-k/2\cdot m)/(k/2) = 2n/k-m$. The area of half polygon area iswould be $S = 1/4((n-(k-1)\cdot m)/k)^2cot(180/k)$$S = 1/8(2n/k-m)^2cot(180/k)$. One needs to find derivative and assign it to zero to find the maximum. Unfortunately I can't do it without computer, but what have to be done is quite clear, may be some one else would like to do it.

  3. Here the strategy must depend strongly on probability distribution of the given time $t$, which is not given [Here I talk about initial formulation of the question, now it is changed].
    For standard uniform distribution from 0 to $T_{max}$, I don't know how to approach it strictly, but the wild guess would be to use logarithmic spiral, like in the task "What is quickest way out from forest, if you lost?".

It is clear that the area must be a polygon with stakes at corners.

I do not have prove, but most probably:

  1. The fastest way to put stake it each corner is to follow perimeter of the polygon. So we must optimise an area between the path and a line (which goes through first and last stake). The maximum area is given by semicircle. (It is the shape of soap bubble put on a table, physical lows there optimise area of the bubble with given volume, which is virtually what we need). The aria would be $S = \pi(\frac{n}{\pi})^2 = n^2/\pi$.

  2. We should use regular polygon here with some number of corners $k$ and side $A = (n-(k-1)\cdot m)/k$. The area is $S = 1/4((n-(k-1)\cdot m)/k)^2cot(180/k)$. One needs to find derivative and assign it to zero to find the maximum. Unfortunately I can't do it without computer, but what have to be done is quite clear, may be some one else would like to do it.

  3. Here the strategy must depend strongly on probability distribution of the given time $t$, which is not given [Here I talk about initial formulation of the question, now it is changed].
    For standard uniform distribution from 0 to $T_{max}$, I don't know how to approach it strictly, but the wild guess would be to use logarithmic spiral, like in the task "What is quickest way out from forest, if you lost?".

It is clear that the area must be a polygon with stakes at corners.

I do not have prove, but most probably:

  1. The fastest way to put stake it each corner is to follow perimeter of the polygon. So we must optimise an area between the path and a line (which goes through first and last stake). The maximum area is given by semicircle. (It is the shape of soap bubble put on a table, physical lows there optimise area of the bubble with given volume, which is virtually what we need). The aria would be $S = \pi(\frac{n}{\pi})^2 = n^2/\pi$.

  2. We should use regular polygon here with some number of corners $k$. But we must follow only half of it, similarly to semicircle (thanks kaine for insight on this). So side of polygon must be $A = (n-k/2\cdot m)/(k/2) = 2n/k-m$. The area of half polygon would be $S = 1/8(2n/k-m)^2cot(180/k)$. One needs to find derivative and assign it to zero to find the maximum. Unfortunately I can't do it without computer, but what have to be done is quite clear, may be some one else would like to do it.

  3. Here the strategy must depend strongly on probability distribution of the given time $t$, which is not given [Here I talk about initial formulation of the question, now it is changed].
    For standard uniform distribution from 0 to $T_{max}$, I don't know how to approach it strictly, but the wild guess would be to use logarithmic spiral, like in the task "What is quickest way out from forest, if you lost?".

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Currently this answer is given in assumption that you must go from last stake to the first. This is not exactly what is asked and I will need to rework it later.

It is clear that the area must be a polygon with stakes at corners.

I do not have prove, but most probably:

  1. The fastest way to put stake it each corner is to follow perimeter of the polygon. ThenSo we must optimise an area between the figure with maximum ariapath and fixed perimeter is circle. The aria would be $S = \pi(\frac{n}{2\pi})^2 = \frac{n^2}{4\pi}$.a line (If you don't have to go fromwhich goes through first and last stake to the first than one can prove that). The maximum area is given by semicircle. (It is optimalthe shape of soap bubble put on a table, physical lows there optimise area of the bubble with given volume, which is virtually what we need). The aria would be $S = \pi(\frac{n}{\pi})^2 = n^2/\pi$.

  2. We should use regular polygon here with some number of corners $k$ and side $A = (n-k\cdot m)/k$$A = (n-(k-1)\cdot m)/k$. The area is $S = 1/4((n-k\cdot m)/k)^2cot(180/k)$$S = 1/4((n-(k-1)\cdot m)/k)^2cot(180/k)$. One needs to find derivative and assign it to zero to find the maximum. Unfortunately I can't do it with outwithout computer, but what have to be done is quite clear, may be some one else would like to do it.

  3. Here the strategy must depend strongly on probability distribution of the given time $t$, which is not given. [Here I talk about initial formulation of the question, now it is changed].
    For standard uniform distribution from 0 to $T_{max}$, I don't know how to approach it strictly, but the wild guess would be to use logarithmic spiral, like in the task "What is quickest way out from forest, if you lost?".

Currently this answer is given in assumption that you must go from last stake to the first. This is not exactly what is asked and I will need to rework it later.

It is clear that the area must be a polygon with stakes at corners.

I do not have prove, but most probably:

  1. The fastest way to put stake it each corner is to follow perimeter of the polygon. Then the figure with maximum aria and fixed perimeter is circle. The aria would be $S = \pi(\frac{n}{2\pi})^2 = \frac{n^2}{4\pi}$. (If you don't have to go from last stake to the first than one can prove that semicircle is optimal).

  2. We should use regular polygon here with some number of corners $k$ and side $A = (n-k\cdot m)/k$. The area is $S = 1/4((n-k\cdot m)/k)^2cot(180/k)$. One needs to find derivative and assign it to zero to find the maximum. Unfortunately I can't do it with out computer.

  3. Here strategy must depend strongly on probability distribution of the given time $t$, which is not given.
    For standard uniform distribution from 0 to $T_{max}$, I don't know how to approach it strictly, but the wild guess would be to use logarithmic spiral, like in the task "What is quickest way out from forest, if you lost?".

It is clear that the area must be a polygon with stakes at corners.

I do not have prove, but most probably:

  1. The fastest way to put stake it each corner is to follow perimeter of the polygon. So we must optimise an area between the path and a line (which goes through first and last stake). The maximum area is given by semicircle. (It is the shape of soap bubble put on a table, physical lows there optimise area of the bubble with given volume, which is virtually what we need). The aria would be $S = \pi(\frac{n}{\pi})^2 = n^2/\pi$.

  2. We should use regular polygon here with some number of corners $k$ and side $A = (n-(k-1)\cdot m)/k$. The area is $S = 1/4((n-(k-1)\cdot m)/k)^2cot(180/k)$. One needs to find derivative and assign it to zero to find the maximum. Unfortunately I can't do it without computer, but what have to be done is quite clear, may be some one else would like to do it.

  3. Here the strategy must depend strongly on probability distribution of the given time $t$, which is not given [Here I talk about initial formulation of the question, now it is changed].
    For standard uniform distribution from 0 to $T_{max}$, I don't know how to approach it strictly, but the wild guess would be to use logarithmic spiral, like in the task "What is quickest way out from forest, if you lost?".

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