Quiz

Zlobny Yashur

Hermit

Zlobny Yashur
After thinking for a bit, I came to the conclusion that it won't be possible to swim perpendicularly to the maniac's movement...
Most likely, the trajectory should be such that the tangent to it at any given time passes through both characters in this problem (the girl will naturally be the point of tangency).;)
An Archimedean spiral or something like that...

Regarding this answer, I can say almost the same thing as I did last time.
The answer cannot be considered satisfactory because it is at least very incomplete and not definitive, and at least some explanations should be given as to what this will lead to.
What does "incomplete" mean? In what sense "not definitive"? I already said what it will lead to:

to an increase in the distance traveled by the maniac

The girl will reach the shore before the maniac reaches the landing point.
Now I'll think about the tangent option.
Added 22 minutes ago
The tangent option might be correct, but it needs to be thought through very carefully...

I'll allow myself to return to the riddle about the girl on the lake and the maniac for a bit longer.
Yashur correctly solved the riddle and proved that if the girl swims in an Archimedean spiral, she will reach the shore before the maniac.
But I was still bothered by one question: was I right to reject his answer as incorrect when Yashur suggested that the girl move along the tangent drawn from the maniac to her??? And now I can confidently say that I was!!! If she swims away from him along the tangent, she will not be caught by him! She won't be caught even if she starts moving at a distance of (3/4)R from the shore and (5/4)R from the maniac - I wrote above how she can achieve this. So this option is even worse than swimming straight to the shore.

At first, I tried to derive the equation of the line along which she would swim away from him along the tangent, but honestly, I couldn't even compose its differential equation. There were some terribly cumbersome expressions. :(
And the question remained open for me. Then I decided to try to simulate this situation: will she be caught by him or not??? And now I can definitely say that if she swims away from him along the tangent, she will definitely not be caught, even if she starts swimming from the initial position at a distance of (5/4)R from him, she still won't be caught, and the curve along which she will swim will not be an Archimedean spiral!!! In general, if anyone is interested, they can look at it themselves and see it for themselves!

While I was writing this post, another thought came to me: let's check what will happen if she swims as Yashur suggested at the very beginning - perpendicular to the maniac's movement. I tried it, and it turned out that no matter where she starts moving, whether from the center of the lake or at a distance of (5/4)R from the maniac, in this case she will simply swim in a circle with a radius equal to a quarter of the pond's radius!!!

In general, the lower the girl's speed, the closer the result will be to the mathematical one, but the longer you will have to wait for it. The most accurate result will be with a girl's speed of 1, but you will have to wait quite a long time. :(

http://www.hermit.besaba.com/e/a/girlandmaniac.zip