5. The lengths of the sides of an isosceles triangle are expressed as natural numbers (in centimeters). The product of the lengths of all sides of this triangle is 2016. What is the perimeter of this triangle?

## Tasks worth 10 points:
First method: Factorize the number 2016 into prime factors:

$2016=2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 7$.

$1 \text{ POINT}$

Since we are dealing with an isosceles triangle, we need to find two sides of equal length and check if all three sides form a triangle (i.e., whether the sum of the lengths of any two sides is strictly greater than the length of the third side). It is sufficient to check whether the sum of the lengths of the legs is greater than the length of the base.

| Leg $b$ | Leg $b$ | Base $a$ | Triangle existence condition |  |
| :--- | :--- | :--- | :---: | :---: |
|  |  |  | $a+b>b$ | $b+b>a$ |
| 1 | 1 | $2016$ | yes | no |
| 2 | 2 | $2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 7=504$ | yes | no |
| $2 \cdot 2=4$ | $2 \cdot 2=4$ | $2 \cdot 3 \cdot 3 \cdot 7=126$ | yes | no |
| $2 \cdot 2 \cdot 3=12$ | $2 \cdot 2 \cdot 3=12$ | $2 \cdot 7=14$ | yes | yes |
| 3 | 3 | $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 7=224$ | yes | no |
| $3 \cdot 2=6$ | $3 \cdot 2=6$ | $2 \cdot 2 \cdot 2 \cdot 7=56$ | yes | no |

(table)

We can conclude that there is only one solution - the triplet 12, 12, 14

NOTE: If all possible combinations of side lengths are listed, but there is no visible or described verification of the triangle inequality, the number of points awarded is reduced by 2. If one or two of the possible combinations of side lengths are missing (with verification of the triangle inequality), the number of points awarded is reduced by 1.

If only the triplet 12, 12, 14 is listed after factorization into prime factors and a range is specified (with verification of the triangle inequality), the task is scored with 3 points, and without verification of the triangle inequality, with 2 points.