Školjka

%V0 Za svaki prirodan broj $n$ određeni su cijeli brojevi $a_n$ i $b_n$ tako da je $$ (1+\sqrt{2})^{2n+1}=a_n+b_n \sqrt{2}.$$ a) Dokažite da su $a_n$ i $b_n$ neparni za svaki $n$. b) Dokažite da je $b_n$ hipotenuza pravokutnog trokuta čije su katete $$ \frac{a_n+(-1)^n}{2}, \ \frac{a_n-(-1)^n}{2}. $$

%V0 Let $n$ be a positive integer and let $x_1\le x_2\le\cdots\le x_n$ be real numbers. Prove that $$\left(\sum_{i,j=1}^{n}|x_i-x_j|\right)^2\le\frac{2(n^2-1)}{3}\sum_{i,j=1}^{n}(x_i-x_j)^2.$$ Show that the equality holds if and only if $x_1, \ldots, x_n$ is an arithmetic sequence.

%V0 Suppose that $x_1, x_2, x_3, \ldots$ are positive real numbers for which $$x^n_n = \sum^{n-1}_{j=0} x^j_n$$ for $n = 1, 2, 3, \ldots$ Prove that $\forall n,$ $$2 - \frac{1}{2^{n-1}} \leq x_n < 2 - \frac{1}{2^n}.$$

%V0 Find all $x,y$ and $z$ in positive integer: $z + y^{2} + x^{3} = xyz$ and $x = \gcd(y,z)$.

%V0 Let $k$ be a positive integer. Show that there are infinitely many perfect squares of the form $n \cdot 2^k - 7$ where $n$ is a positive integer.

%V0 Let $a$, $b$, $c$ be positive real numbers such that $abc = 1$. Prove that $$\frac {1}{a^{3}\left(b + c\right)} + \frac {1}{b^{3}\left(c + a\right)} + \frac {1}{c^{3}\left(a + b\right)}\geq \frac {3}{2}.$$