4. A sequence of 2016 numbers is written. Each one, except the first and the last, is equal to the sum of its neighbors. Find the sum of all 2016 numbers.
Answer: the sum of all numbers is zero.

Solution. Let the first number be $a$, and the second $b$. Denote the third as $x$. Then $a+x=b$ and, therefore, $x=b-a$. Now, express the fourth in terms of $a$ and $b$: $b+y=b-a$ and, therefore, $y=-a$. Continuing, we get: $a, b, b-a, -a, -b, a-b, a, b, \ldots$ This means the numbers repeat with a period of 6. Since 2016:6=336, the sum of all numbers is the sum of the first six, multiplied by 336. The sum of the first six is zero, so the sum of all is zero.

## Grading Criteria.

Only the answer: 0 points.

Specific example analysis: 1 point.