In a sports club, 100 overweight people are training, weighing from 1 to 100 kg. What is the smallest number of teams they can be divided into so that no team has two overweight people, one of whom weighs twice as much as the other?

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Consider all pairs of fat people who should not end up on the same team. Additionally, it can be assumed that all 100 fat people have different weights.

## Solution

Note that the presence of fat people with the same weight only simplifies the problem. Indeed, with each fat person, you can place in the team all fat people of the same weight. In this case, the condition of the problem will still be satisfied. Therefore, we can assume that all 100 fat people have different weights.

In particular, each fat person has no more than one companion who is half as heavy as them, and no more than one who is twice as heavy as them.

Now let's make our fat people hold hands: each of them will offer their left hand to the one who is twice as heavy (if such a person exists), and their right hand to the one who is half as heavy (if such a person exists).

As a result, the entire hundred will break down into non-overlapping "chains." In each chain, the rightmost person is the lightest, and the leftmost person is the heaviest.

We will color each chain in two colors in a checkerboard pattern, starting with the lightest person. Now we will form two desired teams based on the "color" criterion.

## Answer

Into two teams.