Problem 5. A messenger was riding a horse to deliver a message to Ilya Muromets. At some point, he noticed that Ilya Muromets had passed by him (and continued walking in the opposite direction). After 10 seconds (when the horse stopped), the messenger dismounted and ran to catch up with Ilya. How many seconds will it take for the messenger to deliver the message if he runs twice as fast as Ilya Muromets, but five times slower than the galloping horse?
Answer: 110.

Solution. Let Ilya Muromets walk at a speed of $x$ meters per second. Then the messenger's speed is $2x$, and the horse's speed is $10x$. Therefore, 10 seconds after the meeting, the distance between the messenger and the horse will be $10 \cdot x$ (Ilya Muromets has walked) $+10 \cdot 10 x$ (the horse has traveled to a complete stop) $=110 x$. After this, the closing speed of the messenger and Ilya Muromets is $2x-x=x$. Thus, the messenger will deliver the message in 110 seconds.