B3. a) Solve the equations $9^{x}-4 \cdot 5^{2 x-3}=25^{x-1}+2 \cdot 3^{2 x-3}$ and $\sqrt{x+1+\sqrt{x+1+\sqrt{x+3}}}=\sqrt{x+3}$.

b) Write the general form of a polynomial with a leading coefficient of 2, whose roots are equal to the solutions of the equations from a).

## 18th Mathematics Knowledge Competition for Students of Secondary Technical and Vocational Schools  National Competition, April 21, 2018

## Solutions to Problems and Scoring Guide for the First Year

A competitor who arrives at the solution by any correct method (even if the scoring guide does not anticipate it) receives all possible points.

A correct method is considered any procedure that:

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

If an intermediate or final result can be recognized, guessed, read from a diagram, or calculated mentally, the competitor generally receives all the points anticipated. If, however, the solution is guessed (without calculation leading to it) and there is no written verification or proof, it is scored with 0 points.

A competitor who partially solves the problem, showing correct procedures but not clearly leading to the final solution, cannot receive more than half of the possible points.

The asterisk '*' in the points indicates that the point or points can be awarded for a correct procedure, even if the calculation is incorrect.

![](https://cdn.mathpix.com/cropped/2024_06_07_70800c8f4239f8164bacg-18.jpg?height=137&width=220&top_left_y=1551&top_left_x=929)
B3.
Transforming the equation $3^{2 x}-2 \cdot 3^{2 x-3}=5^{2 x-2}+4 \cdot 5^{2 x-3}$ ..... 1 point
Factoring out the common factor ..... $1^{*}$ point
Rearranging to the form $3^{2 x-5}=5^{2 x-5}$ ..... 1 point
Solution of the equation $x=\frac{5}{2}$ ..... 1 point
Solving the radical equation ..... $1^{*}$ point
Rearranging to the form $x^{2}-7 x+6=0$ ..... 1 point
The solution of the equation is $x=1$, the other solution $x=6$ is not valid ..... 1 point
Writing the polynomial $v$ in the general form $p(x)=2 x^{2}-7 x+5$ ..... 1 point