Olympiad problems

1999 Estonia National Olympiad, 1

Let $a, b, c$ and $d$ be non-negative integers. Prove that the numbers $2^a7^b$ and $2^c7^d$ give the same remainder when divided by $15$ iff the numbers $3^a5^b$ and $3^c5^d$ give the same remainder when divided by $16$.

2007 Postal Coaching, 3

Tags: power of 3 , prime , number theory

Suppose $n$ is a natural number such that $4^n + 2^n + 1$ is a prime. Prove that $n = 3^k$ for some nonnegative integer $k$.

1987 Tournament Of Towns, (150) 1

Tags: number theory , digit , last digit , power of 3

Prove that the second last digit of each power of three is even . (V . I . Plachkos)

1998 Czech And Slovak Olympiad IIIA, 4

Tags: max , number theory , power of 3

For each date of year $1998$, we calculate day$^{month}$ −year and determine the greatest power of $3$ that divides it. For example, for April $21$ we get $21^4 - 1998 =192483 = 3^3 \cdot 7129$, which is divisible by $3^3$ and not by $3^4$ . Find all dates for which this power of $3$ is the greatest.