Školjka

%V0 Za koje $n \in \mathbb{N}$ postoji $k\in\mathbb{N}$ takav da $n^k$ ima istu prvu i zadnju znamenku u bazi $10$?

%V0 Imamo različite proste brojeve $p_1$, $p_2$, $\ldots$, $p_{31}$. Dokaži da, ako $30$ dijeli sumu njihovih četvrtih potencija, među njima postoje $3$ uzastopna prosta.

%V0 We call a positive integer $n$ amazing if there exist positive integers $a, b, c$ such that the equality $$n = (b, c)(a, bc) + (c, a)(b, ca) + (a, b)(c, ab)$$ holds. Prove that there exist $2011$ consecutive positive integers which are amazing. Note. By $(m, n)$ we denote the greatest common divisor of positive integers $m$ and $n$.

%V0 We are given a positive integer $n$ which is not a power of two. Show that ther exists a positive integer $m$ with the following two properties: (a) $m$ is the product of two consecutive positive integers; (b) the decimal representation of $m$ consists of two identical blocks with $n$ digits.

%V0 Find all pairs $(m,\,n)$ of integers which satisfy the equation $$(m + n)^4 = m^2n^2 + m^2 + n^2 + 6mn.$$

%V0 Find all positive integers $k$ with the following property: There exists an integer $a$ so that $(a+k)^{3}-a^{3}$ is a multiple of $2007$.