Problem 3. The teacher lined up 20 students from one class and distributed 800 candies among them. Each student was given the task to calculate the quotient $\frac{x}{x+2 k-1}$, where $x$ is the number of candies they received, and $k$ is the ordinal number of their position in the line, counting from left to right. All students received the same quotient. How many candies did the student with the ordinal number 12 receive?
Solution. Let's assume that the first student received $x_{1}$ candies, the second student received $x_{2}$ candies, ..., and the twentieth student received $x_{20}$ candies. According to this, $x_{1}+x_{2}+\cdots+x_{20}=800$. Let $\frac{x_{k}}{x_{k}+2 k-1}=M$, from which $x_{k}=(2 k-1) \frac{M}{1-M}$. Then, for the sum $x_{1}+x_{2}+\cdots+x_{20}$ we get

$$
x_{1}+x_{2}+\cdots+x_{20}=\frac{M}{1-M}(1+3+5+\cdots+39)=\frac{M}{1-M} \frac{40 \cdot 20}{2}=400 \frac{M}{1-M}
$$

From the condition $x_{1}+x_{2}+\cdots+x_{20}=800$, we have $400 \frac{M}{1-M}=800$, i.e., $\frac{M}{1-M}=2$.

According to this, the student who had the 12th rank received $x_{12}=(2 \cdot 12-1) \cdot 2=46$ candies.