B3. Simplify the expression

$$
\frac{\frac{2 x y}{x+y}-x}{\frac{1}{y}+\frac{1}{x-2 y}}+\frac{\left(x^{2}-x y+y^{2}\right)\left(x^{3}-x(x-y)^{2}\right)}{x^{3}+y^{3}}
$$

For which real values of $x$ and $y$ does the expression have no meaning?

## 16th Knowledge Competition

 in mathematics for students of secondary technical and vocational schoolsNational Competition, April 16, 2016

## Problems for 2nd Year

Time for solving: 120 minutes. In section A, we will award three points for each correct answer, and deduct one point for each incorrect answer. Write the answers for section A in the left table, leave the right table blank.

![](https://cdn.mathpix.com/cropped/2024_06_07_7e68135e1a71f8c81168g-06.jpg?height=163&width=319&top_left_y=661&top_left_x=640)

| B1 | B2 | B3 |
| :--- | :--- | :--- |
|  |  |  |
B3.
Expanding fractions to a common denominator $\frac{\frac{2 x y}{x+y}-x}{\frac{1}{y}+\frac{1}{x-2 y}}=\frac{\frac{x y-x^{2}}{x+y}}{\frac{x-y}{y(x-2 y)}}$ ..... 1 point
Resolving the double fraction ..... $1^{*}$ point
Simplifying the first term of the expression $\mathrm{v}-\frac{x y(x-2 y)}{x+y}$ ..... 1 point
Factorization $x^{3}+y^{3}=(x+y)\left(x^{2}-x y+y^{2}\right)$ ..... 1 point
Simplification $\frac{\left(x^{2}-x y+y^{2}\right)\left(x^{3}-x^{3}+2 x^{2} y-x y^{2}\right)}{(x+y)\left(x^{2}-x y+y^{2}\right)}=\frac{x y(2 x-y)}{x+y}$ ..... 1 point
Simplification of the expression $\frac{-x y(x-2 y)}{x+y}+\frac{x y(2 x-y)}{x+y}=x y$ ..... 1 point
The expression is undefined for $y=0, x=-y, x=2 y$ and $x=y$ ..... 2 points
(1 point for two correct observations.)

## Second Year

$$
\begin{array}{|c|c|c|}
\hline \mathrm{A} 1 & \mathrm{~A} 2 & \mathrm{~A} 3 \\
\hline \mathrm{C} & \mathrm{C} & \mathrm{C} \\
\hline
\end{array}
$$