2. (5 points) At the World Meteorologists Conference, each participant in turn announced the average monthly temperature in their hometown. At this moment, all the others recorded the product of the temperatures in their and the speaker's cities. In total, 50 positive and 60 negative numbers were recorded. What is the smallest number of times a positive temperature could have been announced?
Answer: 5

Solution: When solving, we will not consider cities with a zero temperature, as no positive or negative numbers arise from their interactions.

1) Let there be $x$ participants at the conference, each giving $x-1$ answers, so $x(x-1)=60+50$, which means $x=11$.
2) Let there be $y$ people with "positive temperature" at the conference, then there were $(11-y)$ people with "negative temperature". Each "positive" person recorded a positive number $(y-1)$ times, and each "negative" person recorded a negative number $-(10-y)$ times. Then $y(y-1)+(11-y)(10-y)=50 \Rightarrow$ $y^{2}-11 y+30=0 \Rightarrow y=5$ or $y=6$. Both answers satisfy the condition (in both cases, 60 negative numbers are recorded).