Školjka

%V0 Neka su $x,y,z,a,b,c$ cijeli brojevi za koje vrijedi:$$$\begin{align*} x^2+y^2&=a^2\text{,} \\ x^2+z^2&=b^2\text{,} \\ y^2+z^2&=c^2\text{.} \\ \end{align*}$$$ Dokažite da je broj $xyz$ djeljiv s $(a)$ $5$, $(b)$ $55$.

%V0 Dokažite da se prirodan broj može prikazati kao zbroj dva ili više uzastopnih prirodnih brojeva ako i samo ako taj broj nije potencija broja $2$.

%V0 Neka su $a$, $b$, $c$ različiti prirodni brojevi i $k$ prirodan broj takav da vrijedi $$ \displaystyle ab + bc + ca \geqslant 3 k^2 -1 \text{.} $$ Dokaži da je $\displaystyle \frac13\left({a^3 + b^3 + c^3}\right) \geqslant abc + 3k$.

%V0 Let $a$ and $b$ be non-negative integers such that $ab \geq c^2,$ where $c$ is an integer. Prove that there is a number $n$ and integers $x_1, x_2, \ldots, x_n, y_1, y_2, \ldots, y_n$ such that $$\sum^n_{i=1} x^2_i = a, \sum^n_{i=1} y^2_i = b, \text{ and } \sum^n_{i=1} x_iy_i = c.$$

%V0 Let $a_1 \geq a_2 \geq \ldots \geq a_n$ be real numbers such that for all integers $k > 0,$ $$a^k_1 + a^k_2 + \ldots + a^k_n \geq 0.$$ Let $p = max\{|a_1|, \ldots, |a_n|\}.$ Prove that $p = a_1$ and that $$(x - a_1) \cdot (x - a_2) \cdots (x - a_n) \leq x^n - a^n_1$$ for all $x > a_1.$

%V0 Let $a, b, c$ be positive integers satisfying the conditions $b > 2a$ and $c > 2b.$ Show that there exists a real number $\lambda$ with the property that all the three numbers $\lambda a, \lambda b, \lambda c$ have their fractional parts lying in the interval $\left(\frac {1}{3}, \frac {2}{3} \right].$