Školjka

%V0 Neka je $n>1$ neparan cijeli broj pri čemu postoje cijeli brojevi $x_1, x_2, \ldots x_n \geq 0$ koji zadovoljavaju jednadžbe $$ (x_2-x_1)^2+2(x_2+x_1)+1=n^2$$ $$ (x_3-x_2)^2+2(x_3+x_2)+1=n^2$$ $$ \ldots \ldots \ldots \ldots \ldots \ldots \ldots $$ $$ (x_1-x_n)^2+2(x_1+x_n)+1=n^2$$ Pokažite da je ili $x_1=x_n$ ili postoji $j, \ 1 \leq j \leq n-1$ takav da je $x_j=x_{j+1}$.

%V0 Neka je $F_n=x^n \sin nA + y^n \sin nB + z^n \sin nC$, gdje su $x, y, z, A, B, C$ realni brojevi takvi da je $A+B+C=\pi$. Ako je $F_1=F_2=0$, dokažite da je $F_n=0$ za svaki prirodni broj $n$.

%V0 Let $a_{1}, a_{2}...a_{n}$ be non-negative reals, not all zero. Show that that (a) The polynomial $p(x) = x^{n} - a_{1}x^{n - 1} + ... - a_{n - 1}x - a_{n}$ has preceisely 1 positive real root $R$. (b) let $A = \sum_{i = 1}^n a_{i}$ and $B = \sum_{i = 1}^n ia_{i}$. Show that $A^{A} \leq R^{B}$.

%V0 Let $a > 2$ be given, and starting $a_0 = 1, a_1 = a$ define recursively: $$a_{n+1} = \left(\frac{a^2_n}{a^2_{n-1}} - 2 \right) \cdot a_n.$$ Show that for all integers $k > 0,$ we have: $\sum^k_{i = 0} \frac{1}{a_i} < \frac12 \cdot (2 + a - \sqrt{a^2-4}).$

%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 Show that there exists a finite set $A \subset \mathbb{R}^2$ such that for every $X \in A$ there are points $Y_1, Y_2, \ldots, Y_{1993}$ in $A$ such that the distance between $X$ and $Y_i$ is equal to 1, for every $i.$