Hermit
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According to the formulas in this article, if she starts moving north and ends up south, the length of the Archimedean spiral will be k*(f*sqrt(1+f*f)+ln(f+sqrt(1+f*f)))/2 = 2.07918825013928*k
Here, f is the angle, and it is equal to pi/2.
The radius of the lake is, accordingly, k*f, and the circumference of the lake is k*2*pi*f = 9.86960440108936*k.
Yes, indeed, she has to swim less than 4 times the distance the maniac has to run around the entire circle. Okay, Yashchur, you proved everything correctly. Make a wish.
Actually, I saw such a solution myself. Let the radius of the lake be R, the girl's speed be v, and the maniac's speed be 4v. Then the maniac's angular speed around the lake will be 4v/R = v/(R/4). Therefore, at a distance less than R/4 from the center, the girl's angular speed around the center of the lake will be greater, and she, for example, can simply swim towards the maniac from the center at a distance of almost, but less than R/4, and then swim in a circle, not quite reaching the end, and eventually reach a position where she will be on the same line with him, passing through them and the center of the lake, at a distance of (5/4)*R, and accordingly, the shortest distance to the shore for her to swim will be (3/4)R, and the maniac will have to run pi*R to the place where she lands.
(3/4)*R/v < (pi*R)/(4*v), and therefore she will swim to the shore earlier than he runs.
Here, f is the angle, and it is equal to pi/2.
The radius of the lake is, accordingly, k*f, and the circumference of the lake is k*2*pi*f = 9.86960440108936*k.
Yes, indeed, she has to swim less than 4 times the distance the maniac has to run around the entire circle. Okay, Yashchur, you proved everything correctly. Make a wish.
Actually, I saw such a solution myself. Let the radius of the lake be R, the girl's speed be v, and the maniac's speed be 4v. Then the maniac's angular speed around the lake will be 4v/R = v/(R/4). Therefore, at a distance less than R/4 from the center, the girl's angular speed around the center of the lake will be greater, and she, for example, can simply swim towards the maniac from the center at a distance of almost, but less than R/4, and then swim in a circle, not quite reaching the end, and eventually reach a position where she will be on the same line with him, passing through them and the center of the lake, at a distance of (5/4)*R, and accordingly, the shortest distance to the shore for her to swim will be (3/4)R, and the maniac will have to run pi*R to the place where she lands.
(3/4)*R/v < (pi*R)/(4*v), and therefore she will swim to the shore earlier than he runs.
Всё не так плохо как Вы думаете. Всё намного хуже!