1. Solve the equation $1+2^{n}+3^{n}+5^{n}=2 k$ in integers.
# Solution

For $n \geq 1$, the left side becomes an odd number, while $2k$ is even, so there are no solutions.

For $n=0$, we get $k=2$.

For $n=-1$, we get an equation that has no integer solution for $k$.

For $n \leq -2$, the expression on the left takes values greater than 1 but less than 2. Therefore, there is no integer solution for $k$.

Thus, the equation has a unique solution $n=0, k=2$.

## Grading Criteria

Correct answer only - 1 point.

Correct and complete consideration of non-negative values of $n-2$ points.

Correct and complete consideration of negative values of $n-4$ points.

Points are cumulative.