6. Around a circle, 100 integers are written, the sum of which is 1. A chain is defined as several numbers (possibly one) standing consecutively. Find the number of chains, the sum of the numbers in which is positive.
Answer: 4951.

Solution. We will divide all chains (except the chain consisting of all numbers) into pairs that complement each other. If the sum of the numbers in one chain of the pair is $s$, and in the second is $t$, then $s+t=1$. Since $s$ and $t$ are integers, exactly one of them is positive. Therefore, exactly half of all chains have a positive sum. The total number of chains is $100 \cdot 99$ (two complementary chains are determined by the choice of two positions between them). This gives us 4950 chains with a positive sum. Adding to them the chain of all numbers.