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

(6 points)

## You have 120 minutes to solve the problems.

Solve each problem independently on the provided paper, with each problem on a separate page. Write only your code on the sheets. Write your answers in pen, clearly and neatly. Draw graphs with a pencil. If you solve a problem in multiple ways, clearly indicate which solution the evaluator should grade. If you make a mistake, cross it out.

The National Competition Committee wishes you great success.

## dIFA

11th Mathematics Knowledge Competition for Students of Secondary Technical and Vocational Schools National Competition, April 16, 2011

| $\mathrm{N} 1$ | $\mathrm{~N} 2$ | $\mathrm{~N} 3$ | $\mathrm{~N} 4$ | $\mathrm{~N} 5$ |
| :--- | :--- | :--- | :--- | :--- |
|  |  |  |  |  |
5. Squaring $(\sqrt{x}+\sqrt{y})^{2}=x+2 \sqrt{x y}+y$ and factoring the numerator of the fraction $\frac{\sqrt{x} 3-\sqrt{3}}{\sqrt{x}-\sqrt{y}}=\frac{(\sqrt{x}-\sqrt{y})(x+\sqrt{x y}+y)}{\sqrt{x}-\sqrt{y}}$. We simplify the fraction and get $x+\sqrt{x y}+y$. Simplifying the expression $x+2 \sqrt{x y}+y-(x+\sqrt{x y}+y)=\sqrt{x y}$. We rearrange the expression $(\sqrt{x y})^{-1}=\frac{1}{\sqrt{x y}}$. We get $\frac{\sqrt{x y}}{\sqrt{x y}}$, simplify and get 1.

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-09.jpg?height=69&width=1642&top_left_y=888&top_left_x=204)

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-09.jpg?height=65&width=1642&top_left_y=944&top_left_x=207)
Simplifying the fraction $\frac{(\sqrt{x}-\sqrt{y})(x+\sqrt{x y}+y)}{\sqrt{x}-\sqrt{y}}=x+\sqrt{x y}+y \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots 1$ point Simplifying the expression $x+2 \sqrt{x y}+y-(x+\sqrt{x y}+y)=\sqrt{x y} \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \ldots$ point

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-09.jpg?height=71&width=1642&top_left_y=1118&top_left_x=207)

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-09.jpg?height=74&width=1642&top_left_y=1185&top_left_x=207)

## Third Year