5. Given a sequence $x_{n}$ such that $x_{1}=1, x_{2}=2, x_{n+2}=\left|x_{n+1}\right|-x_{n}$. Find $x_{2015}$.
5. Answer. ${ }^{x_{2015}}=0$.

We will show that the given sequence is periodic with a period of 9. Let's find

$x_{1}=1, x_{2}=2, x_{3}=1, x_{4}=-1, x_{5}=0, x_{6}=1, x_{7}=1, x_{8}=0, x_{9}=-1, x_{10}=1, x_{11}=2, \ldots$.

Since the sequence is completely determined by any two consecutive terms, we have obtained that 9 is the period of this sequence. This means that for any $k, r \in N$, the equality $x_{9 k+r}=x_{r}$ holds.

To find $x_{2015}$, we divide the number 2015 by 9 with a remainder: 2015=9$\cdot$223+8. Then $x_{2015}=x_{8}=0$.

## Evaluation Criteria and Organization of Work Checking

Mathematical Olympiad tasks are creative and allow for several different solutions. Additionally, partial progress in problems (for example, considering an important case, proving a lemma, finding an example, etc.) is evaluated. Thus, when calculating the final scores for a problem, the jury takes into account all these cases, as well as possible logical and arithmetic errors in the solutions.

The checking of works at the Mathematical Olympiad is carried out in two stages. In the first stage, the jury checks the works without assigning scores, using a so-called "plus-minus" system. The sign is assigned according to the table provided below. The preliminary evaluation in the "plus-minus" system may be slightly adjusted after discussing the criteria and classifying the cases.

| Correctness of Solution |  |
| :---: | :--- |
| + | Complete correct solution |
| .+ | Correct solution. There are minor flaws that do not significantly affect the solution |
| $\pm$ | The solution is generally correct. However, the solution contains significant errors or cases are omitted, which do not affect the logical reasoning |
| $+/ 2$ | One of two (more complex) significant cases is correctly considered, or in a problem of the type "estimate + example," the estimate is correctly obtained |
| $\mp$ | Auxiliary statements that help in solving the problem are proven |
| .- | Important individual cases are considered in the absence of a solution |
| - | The solution is incorrect, and there is no progress |
| 0 | The solution is absent |

Sometimes the grade "++" is given to note a correct and elegant solution.

After the first stage of checking, the group of checkers for each problem, analyzing and generalizing the provided solutions, highlights various ways of solving, typical partial progress, and main errors. In accordance with the comparative analysis of different progress, a scale of evaluation criteria is developed.

In the second stage, final scores are assigned for each problem. According to the regulations for conducting school mathematical Olympiads, each problem is scored out of 7 points. The table below provides the scale for converting signs into points.

| Sign | + | .+ | $\pm$ | $+/ 2$ | $\mp$ | .- | - | 0 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| Points | 7 | $6-7$ | $5-6$ | 4 | $2-3$ | $0-1$ | 0 | 0 |

The maximum score for completing all tasks is 35.

It is important to note that any correct solution is scored 7 points. Points are not deducted for a solution being too long or for a solution that differs from the one provided in methodological developments or from other solutions known to the jury.

At the same time, any arbitrarily long text of a solution that does not contain useful progress is scored 0 points.

A traditional mistake made by students when solving proof problems is using the statement to be proven as an initial condition or the basis of the proof. For example, in a problem requiring proof that a triangle is isosceles, the proof begins with the words: "Let triangle $A B C$ be isosceles." Such "solutions" are scored 0 points due to a gross logical error.

Each work is evaluated and checked (rechecked) by at least two members of the jury.

After the preliminary results of the Olympiad work checking are published, Participants have the right to review their works, including informing about their disagreement with the assigned scores. In this case, the Chairman of the Olympiad Jury appoints a jury member to reconsider the work. The score for the work can be changed if the Participant's request for a score change is deemed justified. The score change is agreed upon with the Chairman of the Jury and entered into the final table.

A final table is created for each parallel based on the results of the Olympiad.