Školjka

%V0 Let $a$, $b$, $c$ be positive real numbers such that $\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} = a+b+c$. Prove that: $$\frac{1}{(2a+b+c)^2}+\frac{1}{(a+2b+c)^2}+\frac{1}{(a+b+2c)^2}\leq \frac{3}{16}$$ Proposed by Juhan Aru, Estonia

%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 Prove the inequality: $\displaystyle \sum_{i < j}{\frac {a_{i}a_{j}}{a_{i} + a_{j}}}\leq \frac {n}{2(a_{1} + a_{2} + ... + a_{n})}\cdot \sum_{i < j}{a_{i}a_{j}}$ for positive reals $a_{1}$, $a_{2}$, ..., $a_{n}$.

%V0 Let $x_1,x_2,\ldots,x_n$ be arbitrary real numbers. Prove the inequality $$\frac{x_1}{1+x_1^2} + \frac{x_2}{1+x_1^2 + x_2^2} + \cdots + \frac{x_n}{1 + x_1^2 + \cdots + x_n^2} < \sqrt{n}.$$

%V0 Let $a_{1},a_{2},\ldots ,a_{n}$ be positive real numbers such that $a_{1}+a_{2}+\cdots +a_{n}<1$. Prove that $$\frac{a_{1} a_{2} \cdots a_{n} \left[ 1 - (a_{1} + a_{2} + \cdots + a_{n}) \right] }{(a_{1} + a_{2} + \cdots + a_{n})( 1 - a_1)(1 - a_2) \cdots (1 - a_n)} \leqslant \frac{1}{n^{n+1}}$$

%V0 Let $n$ be an integer, $n \geq 3.$ Let $a_1, a_2, \ldots, a_n$ be real numbers such that $2 \leq a_i \leq 3$ for $i = 1, 2, \ldots, n.$ If $s = a_1 + a_2 + \ldots + a_n,$ prove that $$\frac{a^{2}_{1}+a^{2}_{2}-a^{2}_{3}}{a_{1}+a_{2}-a_{3}}+\frac{a^{2}_{2}+a^{2}_{3}-a^{2}_{4}}{a_{2}+a_{3}-a_{4}}+\ldots+\frac{a^{2}_{n}+a^{2}_{1}-a^{2}_{2}}{a_{n}+a_{1}-a_{2}}\leq 2s-2n.$$