B3. Find all real numbers $x$ that satisfy the equation

$$
\sqrt[3]{2 x+13}-\sqrt[3]{2 x-13}=2
$$

## Problems for 3rd Year

Time for solving: 120 minutes. In section A, we will award two points for the correct answer, and deduct one point for an incorrect answer. Write your answers for section A in the left table, leave the right table blank.
![](https://cdn.mathpix.com/cropped/2024_06_07_58dcb28c79ea4cc315e1g-10.jpg?height=172&width=802&top_left_y=648&top_left_x=641)
B3. We raise the equation to the third power, to get

$$
(2 x+13)-3(\sqrt[3]{2 x+13})^{2} \sqrt[3]{2 x-13}+3 \sqrt[3]{2 x+13}(\sqrt[3]{2 x-13})^{2}-(2 x-13)=8
$$

Then we rearrange it

$$
26-3 \sqrt[3]{2 x+13} \sqrt[3]{2 x-13}(\sqrt[3]{2 x+13}-\sqrt[3]{2 x+13})=8
$$

and note that $\sqrt[3]{2 x+13}-\sqrt[3]{2 x+13}=2$, to get

$$
26-6 \sqrt[3]{2 x+13} \sqrt[3]{2 x-13}=8
$$

We rearrange the equation again to $\sqrt[3]{2 x+13} \sqrt[3]{2 x-13}=3$ and raise it to the third power, to get $(2 x+13)(2 x-13)=27$ or $4 x^{2}-169=27$. From this, it follows that $x^{2}=49$ and thus $x= \pm 7$.

Correct raising of the equation to the third power 1 point

Rearrangement of the equation $\mathbf{v} 26-6 \sqrt[3]{2 x+13} \sqrt[3]{2 x-13}=8$ 2 points

Raising of the equation $\sqrt[3]{2 x+13} \sqrt[3]{2 x-13}=3$ to the third power 1 point

Writing the equation $4 x^{2}-169=27$ 1 point Writing both solutions 1 point

2nd method. We rearrange the equation to $\sqrt[3]{2 x+13}=\sqrt[3]{2 x-13}+2$ and then raise it to the third power, to get

$$
2 x+13=2 x-13+6(\sqrt[3]{2 x-13})^{2}+12 \sqrt[3]{2 x-13}+8
$$

We move all terms to one side and simplify to

$$
6(\sqrt[3]{2 x-13})^{2}+12 \sqrt[3]{2 x-13}-18=0
$$

If we introduce a new variable $y=\sqrt[3]{2 x-13}$ and divide the equation by 6, we get the quadratic equation $y^{2}+2 y-3=0$ or $(y+3)(y-1)=0$, which has two solutions, $y=-3$ and $y=1$. From the equation for the new variable, we express $x=\frac{y^{3}+13}{2}$. When $y=-3$, $x=-7$, and when $y=1$, $x=7$. The original equation thus has two solutions, $x=-7$ and $x=7$.
Introduction of a new variable ..... 1 point
Writing the solutions $y_{1}$ and $y_{2}$ ..... 1 point
Writing the solutions of the original equation ..... 2 points
(A contestant gets all points even if they correctly solved the equation without
introducing a new variable.)

## Solutions for 3rd Year

In section A, a correct answer will be worth two points, while an incorrect answer will result in the deduction of one point. To avoid a potential negative final score, each contestant is given an initial 3 points.

| 1 | 2 | 3 |
| :--- | :--- | :--- |
| D | E | A |

Justifications: