Problem 3. Dima lives in a nine-story building. He descends from his floor to the first floor by elevator in 1 minute. Due to his small stature, Dima cannot reach the button for his floor. Therefore, when going up, he presses the button he can reach, and then walks the rest of the way. The entire journey upwards takes 1 minute 10 seconds. The elevator moves up and down at the same speed, and Dima climbs at half the speed of the elevator. On which floor does Dima live?

[6 points] (D.E. Shnol)
Answer. Dima lives on the seventh floor.

First solution. Consider the part of the journey that Dima travels down by elevator and up on foot. On the one hand, the walk takes twice as long, and on the other, it is 10 seconds longer. Therefore, he traveled this part by elevator in 10 seconds and walked it in 20 seconds. Since the entire elevator ride takes 60 seconds, he walked $1 / 6$ of the way.

Notice that he walked a whole number of intervals between floors. Since the building is nine-story, he walked 1 interval and rode 5. Therefore, Dima lives on the 7th floor.

Second solution. Let the elevator move at a speed of $v$ floors per second, Dima lives on the $n$-th floor, and usually gets off on the $m$-th floor. Then

$$
\left\{\begin{array}{l}
\frac{n-1}{v}=60 \\
\frac{m-1}{v}+\frac{n-m}{v} \cdot 2=70
\end{array}\right.
$$

from which $m-1+2 n-2 m=70 v$. Substitute $v=\frac{n-1}{60}$ from the first equation:

$$
\begin{gathered}
2 n-m-1=\frac{n-1}{6} \cdot 7 \\
12 n-6 m-6=7 n-7 \\
5 n+1=6 m
\end{gathered}
$$

Since $m<9$, it is not difficult to find that $n=7$, and $m=6$.