B3. Find all pairs of real numbers $x$ and $y$ that solve the system of equations

$$
\begin{gathered}
\frac{y^{3}+15 x^{2}}{y^{4}-x^{3}}=\frac{y^{2}+15 x}{y^{3}-x^{2}} \\
\frac{1500 y^{3}+4 x^{2}}{9 y^{4}-4}=\frac{1500 y^{2}+4 x}{9 y^{3}-4}
\end{gathered}
$$

## Problems for 2nd Year

Time for solving: 180 minutes. Each problem in set A has exactly one correct answer. In set A, we will award two points for the correct answer, and deduct one point for an incorrect answer. Write the answers for set A in the left table, leave the right table blank. The committee will consider only the answers written in the table for set A.
![](https://cdn.mathpix.com/cropped/2024_06_07_716acb2f660115fb0b44g-06.jpg?height=172&width=808&top_left_y=685&top_left_x=634)
B3. In both equations, we eliminate the fractions and simplify to get

$$
\begin{aligned}
-x^{2} y^{3}+15 x^{2} y^{3} & =-x^{3} y^{2}+15 x y^{4} \\
-6000 y^{3}+36 x^{2} y^{3}-16 x^{2} & =-6000 y^{2}+36 x y^{4}-16 x
\end{aligned}
$$

We rewrite the first equation as \( x^{2} y^{2}(x-y) = -15 x y^{3}(x-y) \) and then as

$$
x y^{2}(x-y)(x+15 y)=0
$$

Thus, the numbers \( x \) and \( y \) satisfy one of the following four possibilities:

1. possibility: \( x=0 \). When we substitute this into the second of the above two equations, we get \( y^{3} = y^{2} \) or \( y^{2}(y-1) = 0 \). Therefore, \( y=0 \) or \( y=1 \). The first possibility is invalid because at \( x=y=0 \), some fractions in the original equations are not well-defined. Thus, only the possibility \( y=1 \) remains.
2. possibility: \( y=0 \). From the second of the above equations, we then get \( x^{2} = x \), so \( x=0 \) or \( x=1 \). The possibility \( x=0 \) is invalid for the same reason as in the previous case, so we get the solution \( x=1 \).
3. possibility: \( y=x \). Substituting this into the second equation, we get \( -6000 x^{3} - 16 x^{2} = -6000 x^{2} - 16 x \), which can be rewritten as \( 6000 x^{2}(1-x) = 16 x(x-1) \), and then as \( 16 x(x-1)(375 x + 1) = 0 \). The possibility \( x=0 \) is invalid because in this case \( y=0 \) and some fractions in the original equations are not well-defined. For the same reason, the possibility \( x=1 \) is also invalid. Therefore, \( 375 x + 1 = 0 \) or \( x = y = -\frac{1}{375} \).
4. possibility: \( x = -15 y \). Substituting this into the second equation and simplifying, we get

$$
8640 y^{5} - 6000 y^{3} - 2400 y^{2} - 240 y = 0
$$

As in the previous case, the possibility \( y=0 \) is invalid because then \( x=0 \). Therefore, we can divide the last equation by \( 240 y \) to get

$$
36 y^{4} - 25 y^{2} + 10 y - 1 = 0
$$

We observe that the last three terms form a perfect square, so we rewrite the equation as \( 36 y^{4} - (5 y - 1)^{2} = 0 \), and then factor the left side using the difference of squares to get

$$
\left(6 y^{2} - 5 y + 1\right)\left(6 y^{2} + 5 y - 1\right) = 0
$$

We factor both terms on the left side of the equation to get

$$
(3 y - 1)(2 y - 1)(6 y - 1)(y + 1) = 0
$$

The solutions are thus \( y = \frac{1}{3} \) and \( x = -5 \), \( y = \frac{1}{2} \) and \( x = -\frac{15}{2} \), \( y = \frac{1}{6} \) and \( x = -\frac{5}{2} \), and \( y = -1 \) and \( x = 15 \). In all cases, all fractions in the original equations are well-defined.

Therefore, all pairs \((x, y)\) that solve the problem are \((0,1)\), \((1,0)\), \(\left(-\frac{1}{375}, -\frac{1}{375}\right)\), \(\left(-5, \frac{1}{3}\right)\), \(\left(-\frac{15}{2}, \frac{1}{2}\right)\), \(\left(-\frac{5}{2}, \frac{1}{6}\right)\), and \((15, -1)\).

A simplified first or second equation, for example: \( x^{3} y^{2} + 14 x^{2} y^{3} - 15 x y^{4} = 0 \), \( 9 x^{2} y^{3} - 9 x y^{4} + 4 x - 4 x^{2} + 1500 y^{2} - 1500 y^{3} = 0 \)

The factored first equation, for example: \( x y^{2}(x-y)(x+15 y) = 0 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ld