<reponame>diego9627/mathOrgConvert
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{\bf \noindent MOP 2008; Josh \\ Red Inequalities}

\be \ii Show that if $a_i$ are reals with $\sum_{i=1}^n a_i = 1$,
then $\sum_{i=1}^n a_i^2 \geq 1/n.$

\ii If $a,b,c>0$, prove that $(a^2b+b^2c+c^2a)(ab^2+bc^2+ca^2)\geq
9a^2b^2c^2.$

\ii Let $(a_i)_{i=1}^n$ be a sequence of positive reals and let
$(b_i)_{i=1}^n$ be a permutation of $(a_i)$. Prove that
$\sum_{i=1}^n a_i/b_i \geq n$.

\ii Let $x,y,z>0$ with $xyz=1$.  Prove that $x+y+z\leq
x^2+y^2+z^2$.
\ii (Romanian selection test) Let $a,b,x,y,z>0$. Prove that
\[\frac x{ay+bz}+\frac y{az+bx} + \frac z{ax+by} \geq \frac 3{a+b}.\]

\ii (IMO 95/2) Let $a, b, c$ be positive real numbers such that
$abc = 1$. Prove that
\[
\frac{1}{a^3(b+c)} + \frac{1}{b^3(c+a)} + \frac{1}{c^3(a+b)}
\geq\frac{3}{2}.
\]

\ii Let $a,b,c,d>0$.  Prove that
\[\frac{1}{a}+\frac{1}{b}+\frac{4}{c} + \frac{16}{d}\geq\frac{64}{a+b+c+d}.\]

\ii (Part of USAMO 79/3) Let $x,y,z\geq 0$ with $x+y+z=1$.  Prove
that
\[x^3+y^3+z^3+6xyz\geq \frac14.\]

\ii (IMO 74/5) Determine all possible values of
\[ S= \frac{a}{a+b+d} +\frac{b}{a + b+c} + \frac{c}{b+c+d} +\frac{d}{a+c+d}\]
where $a,b,c,d$ are arbitrary positive numbers.

\ii (USAMO 77/5) If $a,b,c,d,e$ are positive numbers bounded by
$p$ and $q$, $0<p<q$, prove that
\[(a+b+c+d+e)\Lp\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}\Rp \leq 25+6\Lp\sqrt{\frac{p}{q}}-\sqrt{\frac{q}{p}}\Rp^2,\] and
determine when there is equality.

\ii (IMO 1984/1) Prove that $0\leq yz+zx+xy-2xyz\leq 7/27$, where
$x,y$ and $z$ are non-negative real numbers for which $x+y+z=1$.

\ee
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