Školjka

%V0 Odredi najveću vrijednost realne konstante $\lambda$ takve da za sve pozitivne realne brojeve $u$, $v$, $w$ za koje je $u\sqrt{vw} + v\sqrt{wu} + w\sqrt{uv} \geqslant 1$ vrijedi nejednakost $u + v + w \geqslant \lambda$.

%V0 Let $n \geq 2$ be a fixed integer. Find the least constant $C$ such the inequality $$\sum_{i<j} x_{i}x_{j} \left(x^{2}_{i}+x^{2}_{j} \right) \leq C \left(\sum_{i}x_{i} \right)^4$$ holds for any $x_{1}, \ldots ,x_{n} \geq 0$ (the sum on the left consists of $\binom{n}{2}$ summands). For this constant $C$, characterize the instances of equality.

%V0 (i) If $x$, $y$ and $z$ are three real numbers, all different from $1$, such that $xyz = 1$, then prove that $$\frac {x^{2}}{\left(x - 1\right)^{2}} + \frac {y^{2}}{\left(y - 1\right)^{2}} + \frac {z^{2}}{\left(z - 1\right)^{2}} \geq 1$$. (With the $\sum$ sign for cyclic summation, this inequality could be rewritten as $\sum \frac {x^{2}}{\left(x - 1\right)^{2}} \geq 1$.) (ii) Prove that equality is achieved for infinitely many triples of rational numbers $x$, $y$ and $z$. Author: Walther Janous, Austria

%V0 Neka su $a$, $b$ i $c$ pozitivni realni brojevi takvi da je $abc = 1$. Dokažite: $$a^2 + b^2 + c^2 \geqslant a + b + c$$

%V0 Dani su brojevi $a_1, a_2, \ldots, a_{2012}$ iz intervala $\left[0,1\right]$. Dokaži nejednakost: $$ \left(1-a_1\right)a_2a_3\cdots a_{2012} + a_1\left(1-a_2\right)a_3 \cdots a_{2012} + \cdots + a_1\cdots a_{2011}\left(1-a_{2012}\right) \leq 1 \text{.}$$

%V0 Za realne brojeve $a$ i $b$ vrijedi $a+b = 1$, $a > 0$, $b > 0$. Dokažite da je $$2 < \left(a-\frac{1}{a}\right) \left(b-\frac{1}{b}\right) \leq \frac{9}{4}.$$