B3. Žan wrote down a sequence of four positive real numbers. The first term of the sequence was the number 3, and the last term was the number 9. The first three terms formed a geometric sequence, and the last three terms formed an arithmetic sequence. Determine all four terms of Žan's sequence.

## 57th Mathematical Competition for High School Students in Slovenia

## Regional Competition, April 4, 2013

## Solutions for 1st Year

In section A, a correct answer will be awarded 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 |
| :--- | :--- | :--- |
| $\mathrm{D}$ | $\mathrm{A}$ | $\mathrm{E}$ |

Justifications:
B3. Let the second and third terms of Žan's sequence be denoted by $x$ and $y$. Then $3, x, y$ is a geometric sequence, so $x^{2}=3 y$. At the same time, $x, y, 9$ is an arithmetic sequence, so $2 y=x+9$. From the second equation, we express $y=\frac{x+9}{2}$. When we substitute this into the first equation and rearrange it, we get $2 x^{2}-3 x-27=0$ or $(2 x-9)(x+3)=0$. Since $x$ is positive, $x=\frac{9}{2}$. From this, we also calculate $y=\frac{27}{4}$. Therefore, Žan's sequence is $3, \frac{9}{2}, \frac{27}{4}, 9$.

![](https://cdn.mathpix.com/cropped/2024_06_07_6340f9febbabfa1859cdg-26.jpg?height=63&width=1579&top_left_y=1479&top_left_x=273)

![](https://cdn.mathpix.com/cropped/2024_06_07_6340f9febbabfa1859cdg-26.jpg?height=54&width=1579&top_left_y=1532&top_left_x=273)

![](https://cdn.mathpix.com/cropped/2024_06_07_6340f9febbabfa1859cdg-26.jpg?height=59&width=1579&top_left_y=1581&top_left_x=273)

![](https://cdn.mathpix.com/cropped/2024_06_07_6340f9febbabfa1859cdg-26.jpg?height=48&width=1573&top_left_y=1632&top_left_x=276)

Written sequence ................................................................ 1 point

(If the contestant did not exclude the solution $x=-3$, 1 point is deducted.)


[^0]:    Noting that along with the pair $(m, n)$, the pair $(-m,-n)$ also solves the equation .......... 1 point

    ![](https://cdn.mathpix.com/cropped/2024_06_07_6340f9febbabfa1859cdg-23.jpg?height=49&width=1576&top_left_y=2214&top_left_x=274)

    Solving the quadratic equation in terms of the variable $n$ and calculating the discriminant 1 point Noting that the discriminant must be non-negative and a perfect square .. 1 point

    ![](https://cdn.mathpix.com/cropped/2024_06_07_6340f9febbabfa1859cdg-23.jpg?height=51&width=1576&top_left_y=2413&top_left_x=274)

    All 7 pairs of solutions written down ............................................................................ (If the contestant writes down at least 3 pairs of solutions (and not all solutions), they get 1 point. If the contestant writes down all 7 pairs of solutions, they get 2 points.)