3. Solve the equation: $[20 x+23]=20+23 x$. Recall that $[a]$ denotes the integer part of the number, that is, the greatest integer not exceeding $a$.

(l. S. Koroleva)
Solution. Let both sides of the equation be denoted by $n$, which is an integer. Then $x=\frac{n-20}{23}$ and $n \leqslant 20 x+23<n+1$. This means that

$$
n \leqslant \frac{20(n-20)}{23}+23<n+1
$$

which is

$$
n \leqslant 43<n+7 \frac{2}{3}
$$

It follows that $x=1-\frac{k}{23}$ for any integer $0 \leqslant k \leqslant 7$.

Answer: $\frac{16}{23}, \frac{17}{23}, \frac{18}{23}, \frac{19}{23}, \frac{20}{23}, \frac{21}{23}, \frac{22}{23}, 1$.

Criteria. 0 points - a solution in which it was obtained that $x=1$.

2 points - obtained that $x$ has the form $1-\frac{k}{23}$.

-1 point for each lost or extra answer.