B4. For which positive real numbers $x$ are the values of the expressions $3^{x^{2}-4}, 3^{-2}$, and $9^{-x-\frac{1}{2}}$ consecutive terms of a geometric sequence?

## Solutions and Scoring

A contestant who arrives at the solution by any correct method (even if not anticipated by the scoring) receives all possible points.

For a correct method, each procedure that:

- sensibly takes into account the wording of the problem,
- leads to the solution of the problem,
- is mathematically correct and complete,

is considered.

If an intermediate or final result can be recognized, guessed, read from a diagram, or calculated mentally, the contestant generally receives all the points anticipated. However, if the solution is guessed (and cannot be reached without calculation), such a solution is scored with 0 points.

A contestant who has only partially solved the problem, with correct procedures but no clear path to the final solution, cannot receive more than half of the possible points.

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

## First Year

## Section A

| A1 | A2 | A3 | A4 | A5 | A6 |
| :---: | :---: | :---: | :---: | :---: | :---: |
| A | C | B | D | C | C |
B4. Considering that the quotient of consecutive terms is constant $\frac{a_{2}}{a_{1}}=\frac{a_{3}}{a_{2}}$. We get $\frac{3^{-2}}{3^{2}-4}=\frac{9^{-x-\frac{1}{2}}}{3^{-2}}$. We transform the obtained equation to the form $\frac{1}{81}=3^{x^{2}-4} \cdot 9^{-x-\frac{1}{2}}$ or $3^{-4}=3^{x^{2}-4} \cdot 3^{2\left(-x-\frac{1}{2}\right)}$. We simplify the right side of the equation and get $3^{-4}=3^{x^{2}-2 x-5}$. We equate the exponents $x^{2}-2 x-5=-4$ and solve the resulting quadratic equation $x^{2}-2 x-1=0$. The solutions to the equation are $x_{1,2}=\frac{2 \pm \sqrt{8}}{2}=1 \pm \sqrt{2}$. The solution to the problem is only $x=1+\sqrt{2}$, since $1-\sqrt{2}<0$.

![](https://cdn.mathpix.com/cropped/2024_06_07_d6b556bd14f8001ca6fag-18.jpg?height=83&width=1690&top_left_y=1735&top_left_x=226)

![](https://cdn.mathpix.com/cropped/2024_06_07_d6b556bd14f8001ca6fag-18.jpg?height=48&width=1676&top_left_y=1809&top_left_x=230)

![](https://cdn.mathpix.com/cropped/2024_06_07_d6b556bd14f8001ca6fag-18.jpg?height=57&width=1676&top_left_y=1853&top_left_x=230)

![](https://cdn.mathpix.com/cropped/2024_06_07_d6b556bd14f8001ca6fag-18.jpg?height=52&width=1670&top_left_y=1910&top_left_x=233)

![](https://cdn.mathpix.com/cropped/2024_06_07_d6b556bd14f8001ca6fag-18.jpg?height=63&width=1670&top_left_y=1962&top_left_x=233)

![](https://cdn.mathpix.com/cropped/2024_06_07_d6b556bd14f8001ca6fag-18.jpg?height=63&width=1670&top_left_y=2013&top_left_x=233)