9.4 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.

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# Answer: 4951

Solution. Let's 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.