Školjka

%V0 Odredi sve četveroznamenkaste brojeve $\overline{abcd}$ takve da vrijedi $$ 7 \mid \overline{abcd} \text{,} \quad 7 \mid \overline{dcba} \text{,} \quad 37 \mid \overline{abcd} - \overline{dcba} \text{.}$$

%V0 Determine all integers $k\ge 2$ such that for all pairs $(m,\,n)$ of different positive integers not greater than $k$, the number $n^{n-1}-m^{m-1}$ is not divisible by $k$.

%V0 Dano je $n$ prirodnih brojeva, relativno prostih s $n$. Dokaži da među njima postoji njih nekoliko čiji je umnožak umanjen za $1$ djeljiv s $n$.

%V0 Let $m$ be a positive odd integer, $m > 2.$ Find the smallest positive integer $n$ such that $2^{1989}$ divides $m^n - 1.$

%V0 Prove the inequality a.) $\left( a_{1}+a_{2}+...+a_{k}\right) ^{2}\leq k\left(a_{1}^{2}+a_{2}^{2}+...+a_{k}^{2}\right) ,$ where $k\geq 1$ is a natural number and $a_{1},$ $a_{2},$ $...,$ $a_{k}$ are arbitrary real numbers. b.) Using the inequality (1), show that if the real numbers $a_{1},$ $a_{2},$ $...,$ $a_{n}$ satisfy the inequality $$a_{1}+a_{2}+...+a_{n}\geq \sqrt{\left( n-1\right) \left(a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}\right) },$$ then all of these numbers $a_{1},$ $a_{2},$ $\ldots,$ $a_{n}$ are non-negative.

%V0 Let $a_1, a_2, \ldots, a_n$ be positive real numbers. Prove the inequality $$\binom n2 \sum_{i<j} \frac{1}{a_ia_j} \geq 4 \left( \sum_{i<j} \frac{1}{a_i+a_j} \right)^2$$