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IMO Shortlist 2004 problem A7

Let ${a_1,a_2,\dots,a_n}$ be positive real numbers, ${n>1}$ . Denote by $g_n$ their geometric mean, and by $A_1,\,A_2,\,\dots,\,A_n$ the sequence of arithmetic means defined by
$A_k=\frac{a_1+a_2+\cdots+a_k}{k},\qquad k=1,2,\dots,n.$
Let $G_n$ be the geometric mean of $A_1,A_2,\dots,A_n$ . Prove the inequality $n \root n\of{\frac{G_n}{A_n}}+ \frac{g_n}{G_n}\le n+1$ and establish the cases of equality.

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IMO Shortlist 1995 problem A6

Let $n$ be an integer, $n \geq 3.$ Let $x_1, x_2, \ldots, x_n$ be real numbers such that $x_i < x_{i+1}$ for $1 \leq i \leq n - 1$ . Prove that

$\frac{n(n-1)}{2}\sum_{i < j}x_{i}x_{j}>\left(\sum^{n-1}_{i=1}(n-i)\cdot x_{i}\right)\cdot\left(\sum^{n}_{j=2}(j-1)\cdot x_{j}\right)$

IMO Shortlist 2001 problem A3

Let $x_1,x_2,\ldots,x_n$ be arbitrary real numbers. Prove the inequality

$\frac{x_1}{1+x_1^2} + \frac{x_2}{1+x_1^2 + x_2^2} + \cdots + \frac{x_n}{1 + x_1^2 + \cdots + x_n^2} < \sqrt{n}.$

IMO Shortlist 2006 problem A4

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 $a_{1}$ , $a_{2}$ , ..., $a_{n}$ .

IMO Shortlist 2007 problem A6

Let $a_1, a_2, \ldots, a_{100}$ be nonnegative real numbers such that $a^2_1 + a^2_2 + \ldots + a^2_{100} = 1.$ Prove that
$a^2_1 \cdot a_2 + a^2_2 \cdot a_3 + \ldots + a^2_{100} \cdot a_1 < \frac {12}{25}.$
Author: Marcin Kuzma, Poland

IMO Shortlist 2008 problem A5

Let $a$ , $b$ , $c$ , $d$ be positive real numbers such that $abcd = 1$ and $a + b + c + d > \dfrac{a}{b} + \dfrac{b}{c} + \dfrac{c}{d} + \dfrac{d}{a}$ . Prove that
$a + b + c + d < \dfrac{b}{a} + \dfrac{c}{b} + \dfrac{d}{c} + \dfrac{a}{d}$
Proposed by Pavel Novotný, Slovakia

IMO Shortlist 2008 problem A7

Prove that for any four positive real numbers $a$ , $b$ , $c$ , $d$ the inequality
$\frac {(a - b)(a - c)}{a + b + c} + \frac {(b - c)(b - d)}{b + c + d} + \frac {(c - d)(c - a)}{c + d + a} + \frac {(d - a)(d - b)}{d + a + b} \geqslant 0$
holds. Determine all cases of equality.

Author: Darij Grinberg (Problem Proposal), Christian Reiher (Solution), Germany