First of all, row out to a radius R/4 (where the lake has radius R) keeping you, the centre of the lake and the monster in a straight line - with you on the far side to the monster. This is always possible; radius R/4 is the first point where the angular speed you can achieve just matches that of the monster as he runs round to get you.

You are now a distance 3R/4 away from the shore, directly opposite the monster so he needs to run a distance $pi$ R to get you. You will take time 3R/4V at speed V if you now row directly towards the nearest shore, and he will take $pi$ R/4V, which is fractionally greater.

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For the followup - if instead of 4x, the monster runs Nx your speed... then you row out to radius R/N, you then take time (N-1)R/NV to reach shore and he takes $pi$ R/NV to reach the same point. You escape provided N < $pi$ + 1 = 4.1459

First of all, row out to a radius $R/4$ (where the lake has radius $R$) keeping you, the centre of the lake and the monster in a straight line - with you on the far side to the monster. This is always possible; radius $R/4$ is the first point where the angular speed you can achieve just matches that of the monster as he runs round to get you.

You are now a distance $3R/4$ away from the shore, directly opposite the monster so he needs to run a distance $\pi R$ to get you. You will take time $3R/4V$ at speed $V$ if you now row directly towards the nearest shore, and he will take $\pi R/4V$, which is fractionally greater.

For the followup: If instead of $4\times$, the monster runs $N\times$ your speed... then you row out to radius $R/N$, you then take time $(N-1)R/NV$ to reach shore and he takes $\pi R/NV$ to reach the same point. You escape provided that $N < \pi + 1 \approx 4.1459$.

First of all, row out to a radius R/4 (where the lake has radius R) keeping you, the centre of the lake and the monster in a straight line - with you on the far side to the monster. This is always possible; radius R/4 is the first point where the angular speed you can achieve just matches that of the monster as he runs round to get you.

You are now a distance 3R/4 away from the shore, directly opposite the monster so he needs to run a distance $pi$ R to get you. You will take time 3R/4V at speed V if you now row directly towards the nearest shore, and he will take $pi$ R/4V, which is fractionally greater.

First of all, row out to a radius R/4 (where the lake has radius R) keeping you, the centre of the lake and the monster in a straight line - with you on the far side to the monster. This is always possible; radius R/4 is the first point where the angular speed you can achieve just matches that of the monster as he runs round to get you.

You are now a distance 3R/4 away from the shore, directly opposite the monster so he needs to run a distance $pi$ R to get you. You will take time 3R/4V at speed V if you now row directly towards the nearest shore, and he will take $pi$ R/4V, which is fractionally greater.

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For the followup - if instead of 4x, the monster runs Nx your speed... then you row out to radius R/N, you then take time (N-1)R/NV to reach shore and he takes $pi$ R/NV to reach the same point. You escape provided N < $pi$ + 1 = 4.1459