1. Milan moved 10 steps forward, then returned 2 steps back, then 10 forward and 1 back, then again 10 forward and 2 back, etc. How many steps does Milan need to take to be 1000 steps away from the starting point?
Solution. With 10 steps forward and 2 back, then 10 forward and 1 back, Milan makes 23 steps, but moves 17 steps away. Since

$$
58 \cdot 17=986<59 \cdot 17=1003 \text { steps, }
$$

he will make 58 such cycles in which he will take $58 \cdot 23=1334$ steps. The remaining 14 steps will be taken using $10+2+6=18$ steps. Therefore, Milan will make a total of $1334+18=1356$ steps.