Prove that for all positive real numbers $a,b,c$, \[ \frac{a}{\sqrt{a^2 + 8bc}} + \frac{b}{\sqrt{b^2 + 8ca}} + \frac{c}{\sqrt{c^2 + 8ab}} \geq 1. \]

Given positive numbers ${a_1},{a_2},...,{a_n}$ with ${a_1} + {a_2} + ... + {a_n} = 1$. Prove that $\left( {\frac{1}{{a_1^2}} - 1} \right)\left( {\frac{1}{{a_2^2}} - 1} \right)...\left( {\frac{1}{{a_n^2}} - 1} \right) \geqslant {({n^2} - 1)^n}$.

Prove the inequality [b]a.)[/b] $ \left( a_{1}+a_{2}+...+a_{k}\right) ^{2}\leq k\left( a_{1}^{2}+a_{2}^{2}+...+a_{k}^{2}\right) , $ where $k\geq 1$ is a natural number and $a_{1},$ $a_{2},$ $...,$ $a_{k}$ are arbitrary real numbers. [b]b.)[/b] Using the inequality (1), show that if the real numbers $a_{1},$ $a_{2},$ $...,$ $a_{n}$ satisfy the inequality \[ a_{1}+a_{2}+...+a_{n}\geq \sqrt{\left( n-1\right) \left( a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}\right) }, \] then all of these numbers $a_{1},$ $a_{2},$ $\ldots,$ $a_{n}$ are non-negative.

[list=a] [*]Prove that for any real numbers $a$ and $b$ the following inequality holds: \[ \left( a^{2}+1\right) \left( b^{2}+1\right) +50\geq2\left( 2a+1\right)\left( 3b+1\right)\] [*]Find all positive integers $n$ and $p$ such that: \[ \left( n^{2}+1\right) \left( p^{2}+1\right) +45=2\left( 2n+1\right)\left( 3p+1\right) \][/list]

Let $a_1,a_2,\ldots, a_m$ be a set of $m$ distinct positive even numbers and $b_1,b_2,\ldots,b_n$ be a set of $n$ distinct positive odd numbers such that \[a_1+a_2+\cdots+a_m+b_1+b_2+\cdots+b_n=2019\] Prove that \[5m+12n\le 581.\]

Consider the parabola consisting of the points $(x, y)$ in the real plane satisfying $$(y + x) = (y - x)^2 + 3(y - x) + 3.$$ Find the minimum possible value of $y$.

Let $z_1$ and $z_2$ be complex numbers. Prove that \[|z_1+z_2|+|z_1-z_2|\leqslant |z_1|+|z_2|+\max\{|z_1|,|z_2|\}.\][i]Vlad Cerbu and Sorin Rădulescu[/i]

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

Let $a_1,a_2,\ldots ,a_n$ be $n$ real numbers such that $0<a\le a_k\le b$ for $k=1,2,\ldots ,n$. If $m_1=\frac{1}{n}(a_1+a_2+\cdots+a_n)$ and $m_2=\frac{1}{n}(a_1^2+a_2^2+\cdots + a_n^2)$, prove that $m_2\le\frac{(a+b)^2}{4ab}m_1^2$ and find a necessary and sufficient condition for equality.

To be considered the following complex and distinct $a,b,c,d$. Prove that the following affirmations are equivalent: i)For every $z\in \mathbb{C}$ the inequality takes place :$\left| z-a \right|+\left| z-b \right|\ge \left| z-c \right|+\left| z-d \right|$; ii)There is $t\in \left( 0,1 \right)$ so that $c=ta+\left( 1-t \right)b$ si $d=\left( 1-t \right)a+tb$

Define a function $w:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$ as follows. For $|a|,|b|\le 2,$ let $w(a,b)$ be as in the table shown; otherwise, let $w(a,b)=0.$ \[\begin{array}{|lr|rrrrr|}\hline &&&&b&&\\ &w(a,b)&-2&-1&0&1&2\\ \hline &-2&-1&-2&2&-2&-1\\ &-1&-2&4&-4&4&-2\\ a&0&2&-4&12&-4&2\\ &1&-2&4&-4&4&-2\\ &2&-1&-2&2&-2&-1\\ \hline\end{array}\] For every finite subset $S$ of $\mathbb{Z}\times\mathbb{Z},$ define \[A(S)=\sum_{(\mathbf{s},\mathbf{s'})\in S\times S} w(\mathbf{s}-\mathbf{s'}).\] Prove that if $S$ is any finite nonempty subset of $\mathbb{Z}\times\mathbb{Z},$ then $A(S)>0.$ (For example, if $S=\{(0,1),(0,2),(2,0),(3,1)\},$ then the terms in $A(S)$ are $12,12,12,12,4,4,0,0,0,0,-1,-1,-2,-2,-4,-4.$)

Let $n \ge 2$ be a natural number and suppose that positive numbers $a_0,a_1,...,a_n$ satisfy the equality $(a_{k-1}+a_{k})(a_{k}+a_{k+1})=a_{k-1}-a_{k+1}$ for each $k =1,2,...,n -1$. Prove that $a_n< \frac{1}{n-1}$

Find the minimum value of $S = |x + 1| + |x + 5|+ |x + 14| + |x + 97| + |x + 1920|$.

Find all non-zero real numbers $ x, y, z$ which satisfy the system of equations: \[ (x^2 \plus{} xy \plus{} y^2)(y^2 \plus{} yz \plus{} z^2)(z^2 \plus{} zx \plus{} x^2) \equal{} xyz\] \[ (x^4 \plus{} x^2y^2 \plus{} y^4)(y^4 \plus{} y^2z^2 \plus{} z^4)(z^4 \plus{} z^2x^2 \plus{} x^4) \equal{} x^3y^3z^3\]

For a function $f : [0,1] \to [0,1] $ we define $f^1 = f $ and $f^{n+1} (x) = f (f^n(x))$ for $0 \le x \le 1$ and $n \in N$. Given that there is a $n$ such that $|f^n(x) - f^n(y)| < |x - y| $ for all distinct $x, y \in [0,1]$, prove that there is a unique $x_0 \in [0,1]$ such that $f (x_0) = x_0$.

Let $ n$ be a positive integer and let $ a_i$ be real numbers, $ i \equal{} 1,2,\ldots,n$ such that $ |a_i|\leq 1$ and $ \sum_{i\equal{}1}^n a_i \equal{} 0$. Show that $ \sum_{i\equal{}1}^n |x \minus{} a_i|\leq n$, for every $ x\in \mathbb{R}$ with $ |x|\le 1$.

The set of real numbers $ x$ for which \[ \frac{1}{x\minus{}2009}\plus{}\frac{1}{x\minus{}2010}\plus{}\frac{1}{x\minus{}2011}\ge 1\] is the union of intervals of the form $ a<x\le b$. What is the sum of the lengths of these intervals? $ \textbf{(A)}\ \frac{1003}{335} \qquad \textbf{(B)}\ \frac{1004}{335} \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ \frac{403}{134} \qquad \textbf{(E)}\ \frac{202}{67}$

If $a_1,a_2,\cdots,a_{2013}\in[-2,2]$ and $a_1+a_2+\cdots+a_{2013}=0$ , find the maximum of $a^3_1+a^3_2+\cdots+a^3_{2013}$.

A tree with $n\geq 2$ vertices is given. (A tree is a connected graph without cycles.) The vertices of the tree have real numbers $x_1,x_2,\dots,x_n$ associated with them. Each edge is associated with the product of the two numbers corresponding to the vertices it connects. Let $S$ be a sum of number across all edges. Prove that \[\sqrt{n-1}\left(x_1^2+x_2^2+\dots+x_n^2\right)\geq 2S.\] (Author: V. Dolnikov)

If $0=x_{1}<x_{2}<...<x_{2n+1}=1$ are real numbers with $x_{i+1}-x_{i} \leq h$ for $1 \leq i \leq 2n$, show that $\frac{1-h}{2}<\sum_{i=1}^{n}{x_{2i}(x_{2i+1}-x_{2i-1})}\leq \frac{1+h}{2}$

Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola $ xy\equal{}1$ and both branches of the hyperbola $ xy\equal{}\minus{}1.$ (A set $ S$ in the plane is called [i]convex[/i] if for any two points in $ S$ the line segment connecting them is contained in $ S.$)

Let $n\geqslant 1$ be an integer, and let $x_0,x_1,\ldots,x_{n+1}$ be $n+2$ non-negative real numbers that satisfy $x_ix_{i+1}-x_{i-1}^2\geqslant 1$ for all $i=1,2,\ldots,n.$ Show that \[x_0+x_1+\cdots+x_n+x_{n+1}>\bigg(\frac{2n}{3}\bigg)^{3/2}.\][i]Pakawut Jiradilok and Wijit Yangjit, Thailand[/i]

Let $x_1, ..., x_n$ $(n \geq 2)$ be real numbers from the interval $[1,2]$. Prove that $$|x_1-x_2|+...+|x_n-x_1| + \frac{1}{3} (|x_1-x_3|+...+|x_n-x_2|) \leq \frac{2}{3} (x_1+...+x_n)$$ and determine all cases of equality.

$\boxed{\text{A7}}$ Let $a,b,c$ be positive reals such that $abc=1$.Prove the inequality $\sum\frac{2a^2+\frac{1}{a}}{b+\frac{1}{a}+1}\geq 3$

Problem Shortlist BMO 2017 Let $ a $,$ b$,$ c$, be positive real numbers such that $abc= 1 $. Prove that $$\frac{1}{a^{5}+b^{5}+c^{2}}+\frac{1}{b^{5}+c^{5}+a^{2}}+\frac{1}{c^{5}+b^{5}+b^{2}}\leq 1 . $$