8.1. The teacher suggested that his students - Kolya and Seryozha - solve the same number of problems during the lesson. After some time from the start of the lesson, it turned out that Kolya had solved a third of what Seryozha had left to solve, and Seryozha had left to solve half of what he had already completed. Seryozha calculated that if he continued solving problems at the same speed, he would manage to solve all the problems by the end of the lesson. By what factor does Kolya need to increase his problem-solving speed to solve all the problems by the end of the lesson?
# 8.1. Answer: 16.

Solution. Let Tanya have solved x problems, then she has $\frac{x}{2}$ problems left to solve.

Let $\mathrm{t}_{1}$ be the time interval after which Tanya and Kolya evaluated the shares of solved and remaining problems, and $\mathrm{t}_{2}$ be the remaining time. Since Tanya's problem-solving speed is constant, then $\mathrm{t}_{2}=\mathrm{t}_{1} \cdot \frac{1}{2}$, i.e., $\frac{t_{1}}{t_{2}}=2$.

Then Kolya solved $\frac{1}{3} \cdot \frac{x}{2}=\frac{x}{6}$ problems, and he has $x+\frac{x}{2}-\frac{x}{6}=\frac{8 x}{6}=\frac{4}{3}$ x problems left to solve.

His speed was $\frac{\mathrm{x}}{6 t_{1}}$, and it should become $\frac{4 \mathrm{x}}{3 t_{2}}$.

The ratio of the speeds is $\frac{4 x \cdot 6 t_{1}}{3 t_{2} \cdot x}=\frac{8 t_{1}}{t_{2}}=16$.