Found problems: 10
Let $n \geq 2$ be a fixed positive integer and let $a_{0},a_{1},...,a_{n-1}$ be real numbers. Assume that all of the roots of the polynomial $P(x) = x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+...+a_{1}x+a_{0}$ are strictly positive real numbers. Determine the smallest possible value of $\frac{a_{n-1}^{2}}{a_{n-2}}$ over all such polynomials.
[i]Proposed by Nikola Velov[/i]
Let $n \geq 2$ be an integer and $x_1, x_2, \ldots, x_n$ be positive real numbers such that $\sum_{i=1}^nx_i=1$. Show that $$\bigg(\sum_{i=1}^n\frac{1}{1-x_i}\bigg)\bigg(\sum_{1 \leq i < j \leq n}x_ix_j\bigg) \leq \frac{n}{2}.$$
Let $x,y,z$ be non-negative real numbers such that $xyz=1$. Prove that $$(x^3+2y)(y^3+2z)(z^3+2x) \ge 27.$$
Let $a,b,c,d >0$ and $abcd=1$. Prove that:
\[ \frac{1}{(1+a)^2}+\frac{1}{(1+b)^2}+\frac{1}{(1+c)^2}+\frac{1}{(1+d)^2} \geq 1 \]
Prove that for positive reals $a,b,c$ satisfying $a+b+c=3$ the following inequality holds:
$$ \frac{a}{1+2b^3}+\frac{b}{1+2c^3}+\frac{c}{1+2a^3} \ge 1 $$
Let $a,b,c,d >0$ and $abcd=1$. Prove that:
\[ \frac{1}{(1+a)^2}+\frac{1}{(1+b)^2}+\frac{1}{(1+c)^2}+\frac{1}{(1+d)^2} \geq 1 \]
Prove that for all non-negative numbers $x,y,z$ satisfying $x+y+z=1$, one has
\[1 \le \frac{x}{1-yz}+\frac{y}{1-zx}+\frac{z}{1-xy} \le \frac{9}{8}.\]
Find all real numbers $r$, such that the inequality
\[r(ab+bc+ca)+(3-r)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\geq 9\]
holds for any real $a,b,c>0$.
Let $a,b,c$ be real numbers such that $a^3(b+c)+b^3(a+c)+c^3(a+b)=0$. Prove that $ab+bc+ca\leq0$.
a) Determine the largest real number $A$ with the following property: For all non-negative real numbers $x,y,z$, one has
\[\frac{1+yz}{1+x^2}+\frac{1+zx}{1+y^2}+\frac{1+xy}{1+z^2} \ge A.\]
b) For this real number $A$, find all triples $(x,y,z)$ of non-negative real numbers for which equality holds in the above inequality.