Školjka

%V0 The matrix $$A=\begin{pmatrix} a_{11} & \ldots & a_{1n} \\ \vdots & \ldots & \vdots \\ a_{n1} & \ldots & a_{nn} \end{pmatrix}$$ satisfies the inequality $\sum_{j=1}^n |a_{j1}x_1 + \cdots+ a_{jn}x_n| \leq M$ for each choice of numbers $x_i$ equal to $\pm 1$. Show that $$|a_{11} + a_{22} + \cdots+ a_{nn}| \leq M.$$

%V0 Prove the inequality $$\frac{a_1+ a_3}{a_1 + a_2} + \frac{a_2 + a_4}{a_2 + a_3} + \frac{a_3 + a_1}{a_3 + a_4} + \frac{a_4 + a_2}{a_4 + a_1} \geq 4,$$ where $a_i > 0, i = 1, 2, 3, 4.$

%V0 Let $x_1, x_2, \cdots , x_n$ be positive numbers. Prove that $$\frac{x_{1}^{2}}{x_{1}^{2}+x_{2}x_{3}}+\frac{x_{2}^{2}}{x_{2}^{2}+x_{3}x_{4}}+\cdots+\frac{x_{n-1}^{2}}{x_{n-1}^{2}+x_{n}x_{1}}+\frac{x_{n}^{2}}{x_{n}^{2}+x_{1}x_{2}}\leq n-1$$

%V0 Show that the solution set of the inequality $$\sum^{70}_{k = 1} \frac {k}{x - k} \geq \frac {5}{4}$$ is a union of disjoint intervals, the sum of whose length is 1988.

%V0 Let $w, x, y, z$ are non-negative reals such that $wx + xy + yz + zw = 1$. Show that $$\frac {w^3}{x + y + z} + \frac {x^3}{w + y + z} + \frac {y^3}{w + x + z} + \frac {z^3}{w + x + y}\geq \frac {1}{3}$$.

%V0 Let $n \geq 2, n \in \mathbb{N}$ and let $p, a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n \in \mathbb{R}$ satisfying $\frac{1}{2} \leq p \leq 1,$ $0 \leq a_i,$ $0 \leq b_i \leq p,$ $i = 1, \ldots, n,$ and $$\sum^n_{i=1} a_i = \sum^n_{i=1} b_i.$$ Prove the inequality: $$\sum^n_{i=1} b_i \prod^n_{j = 1, j \neq i} a_j \leq \frac{p}{(n-1)^{n-1}}.$$