1. [4] A hundred friends, including Petya and Vasya, live in several cities. Petya learned the distance from his city to the city of each of the remaining 99 friends and added these 99 numbers. Vasya did the same. Petya got 1000 km. What is the largest number Vasya could have obtained? (Consider the cities as points on a plane; if two people live in the same city, the distance between their cities is considered to be zero.)

Boris Frenkin
Answer: 99000 km.

Solution. Estimate. Let Vasya live in city $V$, and Petya - in city $P$. Consider an arbitrary friend of Vasya (this could be Petya), let's say he lives in city $X$. By the triangle inequality, $V X \leq P V + P X$, and this sum is no more than Petya's total, i.e., 1000 km. Therefore, the sum of the distances from $V$ to the cities of all 99 of Vasya's friends is no more than $99 \cdot 1000$ km.

Example. Everyone except Vasya lives in one city, and Vasya lives 1000 km away from them.