Školjka

%V0 Let $a_1$, $a_2$, $\ldots$, $a_n$ be distinct positive integers, $n\ge 3$. Prove that there exist distinct indices $i$ and $j$ such that $a_i + a_j$ does not divide any of the numbers $3a_1$, $3a_2$, $\ldots$, $3a_n$. Proposed by Mohsen Jamaali, Iran

%V0 A positive integer $N$ is called balanced, if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P\!\left(x\right) = \left(x+a\right)\left(x+b\right)$. a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P\!\left(1\right)$, $P\!\left(2\right)$, ..., $P\!\left(50\right)$ are balanced. b) Prove that if $P\!\left(n\right)$ is balanced for all positive integers $n$, then $a=b$. Proposed by Jorge Tipe, Peru

%V0 Odredite prirodan broj $n$ takav da za njegova četiri najmanja djelitelja $d_1, d_2, d_3, d_4$ vrijedi: $$d_1^2+d_2^2+d_3^2+d_4^2=n.$$

%V0 Za broj kažemo da je [i]malen[/i] ako je strogo manji od zbroja svih svojih djelitelja ne uključujući njega samog. Postoji li [i]malen[/i] neparan broj?

%V0 Find all positive integers $n$ which satisfy the following tow conditions: (a) $n$ has at least four different positive divisors; (b) for any divisors $a$ and $b$ of $n$ satisfying $1<a<b<n$, the number $b-a$ divides $n$.

%V0 Nađite sve prirodne brojeve $x$ i $y$ za koje vrijedi $1! + 2! + 3! + \cdots + x! = y^2$.