1. In Alik's collection, there are two types of items: badges and bracelets. There are more badges than bracelets. Alik noticed that if he increases the number of bracelets by a certain (not necessarily integer) factor, without changing the number of badges, then his collection will have 100 items. And if, conversely, he increases the initial number of badges by the same factor, leaving the number of bracelets unchanged, then he will have 101 items. How many badges and how many bracelets could Alik have in his collection?
Solution. Let Alik have $x$ badges and $y$ bracelets, and the increase occurs by a factor of $n$. Then we get the system

$$
x+n y=100, \quad n x+y=101
$$

By adding and subtracting the equations and eliminating $n$, we obtain

$$
\frac{201}{x+y}-\frac{1}{x-y}=2
$$

We transform this equation to the form

$$
(201-2 u)(2 v+1)=201=3 \cdot 67
$$

where $u=x+y, v=x-y$ are natural numbers.

There are two cases:

1) $2 v+1=3,201-2 u=67$, then $x=34, y=33, n=2$.
2) $2 v+1=67,201-2 u=3$, then $x=66, y=33, n=\frac{34}{33}$.

Answer. 34 badges and 33 bracelets, or 66 badges and 33 bracelets.