Problem 2. The teacher wrote two numbers on the board. Petya divided the first number by the second. Vasya added both numbers and divided the resulting sum by the doubled first number. It turned out that Petya and Vasya got the same result, which is not equal to 1. What is this result?
Answer: -1/2. Solution: Let the first number be $a$, the second number be $b$. According to the condition, $a / b = (a + b) / 2a$. Multiplying both sides of this equation by $2ab$, we get $2a^2 = ab + b^2 \Rightarrow a^2 - b^2 + a^2 - ab = 0 \Rightarrow (a - b)(2a + b) = 0$. Note that $a \neq b$, since according to the condition $a / b \neq 1$. Therefore, $a - b \neq 0$, from which $2a + b = 0 \Rightarrow b = -2a \Rightarrow a / b = -1 / 2$.

- Correctly setting up the equation according to the problem statement, but no further progress - 0 points. Dividing by $a - b$ without justification that $a \neq b$ - the solution is not rated higher than 4 points.