[Examples and counterexamples. Constructions]

A snail woke up, crawled from the mushroom to the spring, and fell asleep. The journey took six hours. The snail moved sometimes faster, sometimes slower, and stopped. Several scientists observed the snail. It is known that:

1) At every moment of the journey, the snail was observed by at least one scientist.
2) Each scientist observed the awake snail for one hour (continuously) and says that during this time the snail crawled exactly one meter.

What is the maximum possible distance from the mushroom to the spring?
Example. Let's divide the entire time of the snail's movement into 30 intervals of 12 minutes each. Suppose the snail crawls 1 m in each of the 10 intervals numbered $1, 6, 7, 12, 13, 18, 19, 24, 25$, and 30, and rests the rest of the time. Scientists observe it in intervals 1-5 (from 1st to 5th - exactly one hour), 2-6, 7-11, 8-12, 13-17, 14-18, 19-23, $20-24, 25-29, 26-30$.

See the estimate in problem 73652.

## Answer

$10 \text{ m}$.
a) 2000 chips are placed on a plane at the vertices of a convex 2000-gon. In one move, you can divide them into two groups and move the chips of the first group by some vector, leaving the other chips in place. Can it happen that after 9 moves all the chips end up on one line? b) And after 10 moves?

## Hint

If after the $\mathrm{n}$-th move the chips lie on $\mathrm{k}$ lines, then after the (n-1)-th move they must have lain on $2 \mathrm{k}$ lines.

## Solution

a) If after the n-th move the chips are on one line, then after the (n-1)-th move they must have been on two lines, after the (n-2)-th - on 4 lines, and so on, and initially - on $2^{\text {n }}$ lines. Since no three vertices of a convex polygon lie on one line, the smallest number of lines containing all vertices of a 2000-gon is 1000. Therefore, $2^{\mathrm{n}}$ is not less than 1000, which means n is not less than 10, solving part a). b) Consider 1024 parallel lines, the distances between adjacent ones of which are 1. Place the vertices of the convex 2000-gon on these lines. The vectors of displacement will be perpendicular to the lines, and we will use each move to merge pairs of adjacent lines so that after 10 moves only one line remains.