1. The pond has a rectangular shape. On the first frosty day, the part of the pond within 10 meters of the nearest shore froze. On the second day, the part within 20 meters froze, on the third day, the part within 30 meters, and so on. On the first day, the area of open water decreased by 20.2%, and on the second day, it decreased by 18.6% of the original area. On which day will the pond be completely frozen?
Solution. First method. Let the sides of the pond be $a$ and $b$ meters, then

$(a-20)(b-20)=(1-0.202) a b, (a-40)(b-40)=(1-0.388) a b$,

from which $20(a+b)-400=0.202 a b, 40(a+b)-1600=0.388 a b$, that is, $800=0.016 a b, a b=5000$ and further $a+b=525$. It turns out that the sides are 400 and 125 meters.

Answer: on the seventh day

![](https://cdn.mathpix.com/cropped/2024_05_06_2a8605c0f478f14359cbg-1.jpg?height=385&width=800&top_left_y=821&top_left_x=695)

Second method. Notice that each day 800 m $^{2}$ less freezes than the previous day. This is evident from the diagram, which shows that the "outer frame" consists of pieces equal to the corresponding pieces of the "inner frame," plus eight 10 x 10 m squares. Therefore, the percentage of the frozen part also decreases by the same amount each day. That is, on the first day, 20.2% of the area freezes, on the second day 18.6%, on the third day 17.0%, and so on. Note that the sum of the first six terms of this progression is less than 100%, while the sum of the first seven terms is already more than 100%. Therefore, the pond will freeze on the seventh day.