Problem 1. Vasya and Petya ran out from the starting point of a circular running track at the same time and ran in opposite directions. They met at some point on the track. Vasya ran a full lap and, continuing to run in the same direction, reached the place of their previous meeting at the moment when Petya had run a full lap. How many times faster did Vasya run than Petya?
Answer: $\frac{\sqrt{5}+1}{2}$.

Solution. Let $v-$ be Petya's speed, $x v$ - Vasya's speed, $t$ - the time it took for them to reach the meeting point. Then from the condition we have the equation $\frac{(1+x) v t}{x v}=\frac{x v t}{v}$, from which $x^{2}=1+x, x=\frac{\sqrt{5}+1}{2}$.

Answer to variant 2: $\frac{\sqrt{5}+1}{2}$.