2. Is it possible to choose natural coefficients $A, B$, and $C$ in the expression $A \cdot 5^{n}+B \cdot 3^{n-1}+C$ so that none of them is divisible by 8, but the result is divisible by 8 for any natural $n$?

(L. S. Korechkova)
Solution. Yes. Note that $5^{n}$ gives a remainder of 1 when divided by 4, and $3^{n-1}$ is always odd. Therefore, we can take $A=2, B=4, C=2$.

Criteria. Answer "yes" without explanation - 0 points. No more than one point if one of the coefficients is a multiple of 8.

Correct coefficients given without explanation - 3 points; correct coefficients and an attempt to prove that they fit, not fully completed (for example, incomplete enumeration) $-\approx 5$ points.