10-6. The places of cyclists in the race are determined by the sum of the times on all stages: the first place goes to the rider with the smallest total time,..., the last place goes to the rider with the largest total time. There were 500 cyclists, the race took place over 15 stages, and there were no cyclists with the same times either on the stages or in the total time over all stages. Vasya always finished seventh. What is the lowest place (i.e., the place with the highest number) he could have taken?
Answer: 91.

Solution. Estimation. Note that if driver A has a higher position than driver B, then A must have overtaken B in at least one race. Vasya was overtaken by no more than $6 \cdot 15=90$ other drivers in 15 races (no more than because some drivers could have overtaken Vasya in several races). Therefore, there can be no more than 90 drivers above Vasya, which means Vasya could not have finished lower than 91st place.

Example. We will show that Vasya could indeed have finished in 91st place. Suppose that in each of the 15 races, 7 different drivers overtook Vasya, and each time they were different. They finished approximately in 5 hours, Vasya in exactly 10 hours, and the rest in approximately 10.1 hours.

Then the total time for each of the 90 drivers who overtook Vasya is approximately $5+14 \cdot 10.1=146.4$ hours, Vasya's total time is $15 \cdot 10=150$ hours, and the total time for each of the other drivers is $15 \cdot 10.1=151.5$ hours. Vasya is in 91st place. It remains to adjust the exact times (instead of "approximately 5 hours," "approximately 10.1 hours") so that no drivers have the same position. It is clear that this can be done by adding fractions of a second to some drivers in some races: all times that should be different will be made different, but Vasya will remain in 91st place.