B4 For digging a well, we pay 60 euros for the first meter of excavation, and for each subsequent meter, we pay 6 euros more. Calculate the depth of the well if the excavation cost is 870 euros. How much do we pay for the last meter of the well excavation?

Space for solving the problems in set $B$.

## Solutions and Scoring

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

A correct method is one 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 contestant generally receives all anticipated points. However, if the solution is guessed (if it cannot be arrived at without calculation), such a solution is scored with 0 points.

If a contestant has only partially solved the problem, from otherwise correct solving procedures but the path to the final solution of the problem is not visible, the contestant cannot receive more than half of the possible points. To avoid a possible negative final score, each contestant is credited with an initial 3 points.

## First Year

## Set A

| A1 | A2 | A3 | A4 | A5 | A6 |
| :---: | :---: | :---: | :---: | :---: | :---: |
| B | B | B | A | A | E |
B4. We use the formula $s_{n}=\frac{n}{2}\left(2 a_{1}+(n-1) d\right)$ and get $870=\frac{n}{2}(120+(n-1) 6)$. We simplify the equation, divide by 3, and get $n^{2}+19 n-290=0$. We factor the equation $(n-10)(n+29)=0$. The appropriate solution to the equation is $n=10$, which means the depth of the well is $10 \mathrm{~m}$. We also calculate the tenth term of the arithmetic sequence for the cost of the last meter of excavation $a_{10}=a_{1}+9 d$ and get $a_{10}=60+54=114$. We pay 114 euros for the last meter of excavation.
Writing the equation $870=\frac{n}{2}(120+(n-1) 6)$ ..... 1 point
Simplifying the equation $n^{2}+19 n-290=0$ ..... 1 point
Solving the equation ..... $1^{*}$ point
Solution $n=10$ ..... 1 point
Calculating the tenth term of the sequence $a_{10}=60+54=114$ ..... 1 point
Writing the answer ..... 1 point
NOTE: If the depth of the well is correctly calculated without using formulas and without the answer, 3 points are awarded.