For $p>1,\frac1p+\frac1q=1$ and $r>1$. If $x_{00},y_{00}>0$, and reals $x_{ij},y_{ij},i=1,2,\ldots,n$, $j=1,2,\ldots,m$, then prove that $$\left(\frac{\left(\displaystyle\sum_{j=1}^m\displaystyle\sum_{i=1}^n(x_{ij}+y_{ij})^r\right)^{1/r}}{(x_{00}+y_{00})^{1/q}}\right)^p\le\left(\frac{\left(\displaystyle\sum_{j=1}^m\displaystyle\sum_{i=1}^nx_{ij}^r\right)^{1/r}}{x_{00}^{1/q}}\right)^p+\left(\frac{\left(\displaystyle\sum_{j=1}^m\displaystyle\sum_{i=1}^ny_{ij}^r\right)^{1/r}}{y_{00}^{1/q}}\right)^p$$ with equality if and only if either $x_{ij}=y_{ij}=0$ for $i=1,\ldots,n,j=1,\ldots,m$ or $x_{ij}=\alpha y_{ij}$ for $i=0,1,\ldots,n,j=0,1,\ldots,m$, and some $\alpha>0$. [i]Proposed by Chang-Jian Zhao[/i]

In the $ ABC$ triangle, the bisector of $A $ intersects the $ [BC] $ at the point $ A_ {1} $ , and the circle circumscribed to the triangle $ ABC $ at the point $ A_ {2} $. Similarly are defined $ B_ {1} $ and $ B_ {2} $ , as well as $ C_ {1} $ and $ C_ {2} $. Prove that $$ \frac {A_{1}A_{2}}{BA_{2} + A_{2}C} + \frac {B_{1}B_{2}}{CB_{2} + B_{2}A} + \frac {C_{1}C_{2}}{AC_{2} + C_{2}B} \geq \frac {3}{4}$$

Prove that for all non-negative numbers $x,y,z$ satisfying $x+y+z=1$, one has \[1 \le \frac{x}{1-yz}+\frac{y}{1-zx}+\frac{z}{1-xy} \le \frac{9}{8}.\]

Prove that for every numbers $a,b,c>0$ the following inequality is true $$\frac{a^4-a^2+1}{b^5}+\frac{b^4-b^2+1}{c^5}+\frac{c^4-c^2+1}{a^5} \geq \frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}.$$

$a_1,…,a_n$ are real numbers such that $a_1+…+a_n=0$ and $|a_1|+…+|a_n|=1$. Prove that : $$|a_1+2a_2+…+na_n| \leq \frac{n-1}{2}$$

Prove for all positive reals a,b,c,d: $ \frac{a\minus{}b}{b\plus{}c}\plus{}\frac{b\minus{}c}{c\plus{}d}\plus{}\frac{c\minus{}d}{d\plus{}a}\plus{}\frac{d\minus{}a}{a\plus{}b} \geq 0$

Find the smallest value of the expression: $$y=\frac{x^2}{8}+x \cos x +\cos 2x$$

Let $x,y$ be positive real numbers such that $x+y=1$. Prove that $\left(1+\frac {1}{x}\right)\left(1+\frac {1}{y}\right)\geq 9$.

Let $a,b,c$ be positive real numbers for which $ab+bc+ca=\frac{1}{3}$. Prove the inequality \[ \frac{a}{a^2-bc+1}+\frac{b}{b^2-ca+1}+\frac{c}{c^2-ab+1}\ge\frac{1}{a+b+c}\]

The product of positive numbers $x, y$ and $z$ is equal to $1$. Prove that if it holds that $$\frac1x +\frac1y + \frac1z \ge x + y + z,$$ then for any natural $k$, holds the inequality $$\frac{1}{x^k} +\frac{1}{y^k} + \frac{1}{z^k} \ge x^k + y^k + z^k.$$

prove that for almost every real number $\alpha \in [0,1]$ there exists natural number $n_{\alpha} \in \mathbb N$ such that the inequality $|\alpha-\frac{p}{q}|\le \frac{1}{q^n}$ for natural $n\ge n_{\alpha}$ and rational $\frac{p}{q}$ has no answers.

Four real numbers $ p$, $ q$, $ r$, $ s$ satisfy $ p+q+r+s = 9$ and $ p^{2}+q^{2}+r^{2}+s^{2}= 21$. Prove that there exists a permutation $ \left(a,b,c,d\right)$ of $ \left(p,q,r,s\right)$ such that $ ab-cd \geq 2$.

Let $ n$ be an integer,$ n \geq 3.$ Let $ x_1, x_2, \ldots, x_n$ be real numbers such that $ x_i < x_{i\plus{}1}$ for $ 1 \leq i \leq n \minus{} 1$. Prove that \[ \frac{n(n\minus{}1)}{2} \sum_{i < j} x_ix_j > \left(\sum^{n\minus{}1}_{i\equal{}1} (n\minus{}i)\cdot x_i \right) \cdot \left(\sum^{n}_{j\equal{}2} (j\minus{}1) \cdot x_j \right)\]

Find all $a,b,c \in \mathbb{N}$ such that \[a^2b|a^3+b^3+c^3,\qquad b^2c|a^3+b^3+c^3, \qquad c^2a|a^3+b^3+c^3.\] [PS: The original problem was this: Find all $a,b,c \in \mathbb{N}$ such that \[a^2b|a^3+b^3+c^3,\qquad b^2c|a^3+b^3+c^3, \qquad \color{red}{c^2b}|a^3+b^3+c^3.\] But I think the author meant $c^2a|a^3+b^3+c^3$, just because of symmetry]

Prove that \[ 1-\frac{1}{2012}\left(\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{2013}\right)\ge \frac{1}{\sqrt[2012]{2013}}.\]

Which is greater: $ 17091982!^2$ or $ 17091982^{17091982}$?

Let $n \geq 2$ be an integer. Positive reals satisfy $a_1\geq a_2\geq \cdots\geq a_n.$ Prove that $$\left(\sum_{i=1}^n\frac{a_i}{a_{i+1}}\right)-n \leq \frac{1}{2a_1a_n}\sum_{i=1}^n(a_i-a_{i+1})^2,$$ where $a_{n+1}=a_1.$

Find all positive integers $n$ which satisfy the following tow conditions: (a) $n$ has at least four different positive divisors; (b) for any divisors $a$ and $b$ of $n$ satisfying $1<a<b<n$, the number $b-a$ divides $n$. [i](4th Middle European Mathematical Olympiad, Individual Competition, Problem 4)[/i]

Find all pairs of positive integers $ (m,n)$ such that $ mn \minus{} 1$ divides $ (n^2 \minus{} n \plus{} 1)^2$. [i]Aaron Pixton.[/i]

Find the largest positive integer $n>10$ such that the residue of $n$ when divided by each perfect square between $2$ and $\dfrac n2$ is an odd number.

Show that for all non-negative numbers $a,b$, $$ 1 + a^{2017} + b^{2017} \geq a^{10}b^{7} + a^{7}b^{2000} + a^{2000}b^{10} $$When is equality attained?

Solve the inequality $$3-2\left(3-2\left(3-...-2(3-2x)\right)...\right) >x$$. The total number of right parentheses is $100$.

Let $ n \ge 2$ be a fixed integer and let $ a_{i,j} (1 \le i < j \le n)$ be some positive integers. For a sequence $ x_1, ... , x_n$ of reals, let $ K(x_1, .... , x_n)$ be the product of all expressions $ (x_i \minus{} x_j)^{a_{i,j}}$ where $ 1 \le i < j \le n$. Prove that if the inequality $ K(x_1, .... , x_n) \ge 0$ holds independently of the choice of the sequence $ x_1, ... , x_n$ then all integers $ a_{i,j}$ are even.

The minimal value of $ f(x) \equal{} \sqrt{a^2 \plus{} x^2} \plus{} \sqrt{(x\minus{}b)^2 \plus{} c^2}$ is A. $ a\plus{}b\plus{}c$ B. $ \sqrt{a^2 \plus{} (b \plus{} c)^2}$ C. $ \sqrt{b^2 \plus{} (a\plus{}c)^2}$ D. $ \sqrt{(a\plus{}b)^2 \plus{} c^2}$ E. None of these

Let $a$, $b$ and $c$ be any positive real numbers. Prove that $$\dfrac{a^2+b^2}{2c}+\dfrac{a^2+c^2}{2b}+\dfrac{b^2+c^2}{2a} \geqslant a+b+c.$$