Let $a,b,c$ be positive real numbers such that $a+b+c = 3$. Find the minimum value of the expression \[A=\dfrac{2-a^3}a+\dfrac{2-b^3}b+\dfrac{2-c^3}c.\]

a) For a finite set of natural numbers $S$, denote by $S+S$ the set of numbers $z=x+y$, where $x,y\in S$. Let $m=|S|$. Show that $|S+S|\leq \frac{m(m+1)}{2}$. b) Let $m$ be a fixed positive integer. Denote by $C(m)$ the greatest integer $k\geq 1$ for which there exists a set $S$ of $m$ integers, such that $\{1,2,\ldots,k\}\subseteq S\cup(S+S)$. For example, $C(3)=8$, with $S=\{1,3,4\}$. Show that $\frac{m(m+6)}{4}\leq C(m) \leq \frac{m(m+3)}{2}$.

Suppose $ a, b,$ and $ c$ are positive integers with $ a \plus{} b \plus{} c \equal{} 2006$, and $ a!b!c! \equal{} m\cdot10^n$, where $ m$ and $ n$ are integers and $ m$ is not divisible by 10. What is the smallest possible value of $ n$? $ \textbf{(A) } 489 \qquad \textbf{(B) } 492 \qquad \textbf{(C) } 495 \qquad \textbf{(D) } 498 \qquad \textbf{(E) } 501$

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]

For any real number$m$, the equation $$x^2+(m-2)x- (m+3)=0$$ has two solutions, denoted $x_1 $and $ x_2$. Determine $m$ such that $x_1^2+x_2^2$ is the minimum possible.

Let $a,b,c$ be positive real numbers . Prove that$$ \frac{1}{ab(b+1)(c+1)}+\frac{1}{bc(c+1)(a+1)}+\frac{1}{ca(a+1)(b+1)}\geq\frac{3}{(1+abc)^2}.$$

The product of three positive numbers is $1$ and their sum is greater than the sum of their inverses. Prove that one of these numbers is greater than $1$, while the other two are smaller than $1$.

a) Let $a$ and $b$ be two non-negative real numbers. Show that $a+b \ge \sqrt{\frac{a^2+b^2}{2}}+ \sqrt{ab}$ b) Let $a$ and $b$ be two real numbers in $[0, 3]$. Show that $\sqrt{\frac{a^2+b^2}{2}}+ \sqrt{ab} \ge \frac{(a+b)^2}{2}$

Let $ABCDEF$ be a convex hexagon such that $AB=BC,CD=DE, EF=FA.$ Prove that $\frac{BC}{BE}+\frac{DE}{DA}+\frac{FA}{FC}\ge\frac{3}{2}$ and find when equality holds.

a) Prove that for $x,y \ge 1$, holds $$x+y - \frac{1}{x}- \frac{1}{y} \ge 2\sqrt{xy} -\frac{2}{\sqrt{xy}}$$ b) Prove that for $a,b,c,d \ge 1$ with $abcd=16$ , holds $$a+b+c+d-\frac{1}{a}-\frac{1}{b}-\frac{1}{c}-\frac{1}{d}\ge 6$$

Let $ a,b,c,d$ be real numbers, such that $ a^2\le 1, a^2 \plus{} b^2\le 5, a^2 \plus{} b^2 \plus{} c^2\le 14, a^2 \plus{} b^2 \plus{} c^2 \plus{} d^2\le 30$. Prove that $ a \plus{} b \plus{} c \plus{} d\le 10$.

For a positive integer $k$, let us denote by $u(k)$ the greatest odd divisor of $k$. Prove that, for each $n \in N$, $\frac{1}{2^n} \sum_{k = 1}^{2^n} \frac{u(k)}{k}> \frac{2}{3}$.

Let $n$ be a positive integer. There are given $n$ lines such that no two are parallel and no three meet at a single point. a) Prove that there exists a line such that the number of intersection points of these $n$ lines on both of its sides is at least $$\left \lfloor \frac{(n-1)(n-2)}{10} \right \rfloor.$$ Notice that the points on the line are not counted. b) Find all $n$ for which there exists a configurations where the equality is achieved.

The set $\{1, 2, \cdots, 49\}$ is divided into three subsets. Prove that at least one of these subsets contains three different numbers $a, b, c$ such that $a + b = c$.

Prove that $$x^2 +\frac{8}{xy}+ y^2 \ge 8$$ for all positive real numbers $x$ and $y$.

Let $ ABC$ a triangle and $ X$, $ Y$ and $ Z$ points at the segments $ BC$, $ AC$ and $ AB$, respectively.Let $ A'$, $ B'$ and $ C'$ the circuncenters of triangles $ AZY$,$ BXZ$,$ CYX$, respectively.Prove that $ 4(A'B'C')\geq(ABC)$ with equality if and only if $ AA'$, $ BB'$ and $ CC'$ are concurrents. Note: $ (XYZ)$ denotes the area of $ XYZ$

Let $ \theta_1, \theta_2,\ldots , \theta_{2008}$ be real numbers. Find the maximum value of $ \sin\theta_1\cos\theta_2 \plus{} \sin\theta_2\cos\theta_3 \plus{} \ldots \plus{} \sin\theta_{2007}\cos\theta_{2008} \plus{} \sin\theta_{2008}\cos\theta_1$

Let $a,b,c,d,x,y$ denote the lengths of the sides $AB, BC,CD,DA$ and the diagonals $AC,BD$ of a cyclic quadrilateral $ABCD$ respectively. Prove that $$(\frac{1}{a}+\frac{1}{c})^2+(\frac{1}{b}+\frac{1}{d})^2 \geq 8 ( \frac{1}{x^2}+\frac{1}{y^2})$$

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]

If $a$, $b$ ,$c$ are three positive real numbers such that $ab+bc+ca = 1$, prove that \[ \sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}. \]

Let $a_{1}, a_{2}, a_{3}, ... a_{2011}$ be nonnegative reals with sum $\frac{2011}{2}$, prove : $|\prod_{cyc} (a_{n} - a_{n+1})| = |(a_{1} - a_{2})(a_{2} - a_{3})...(a_{2011}-a_{1})| \le \frac{3 \sqrt3}{16}.$

Determine the number of real solutions of the system \[\left\{ \begin{aligned}\cos x_{1}&= x_{2}\\ &\cdots \\ \cos x_{n-1}&= x_{n}\\ \cos x_{n}&= x_{1}\\ \end{aligned}\right.\]

If $a_1, a_2, \ldots , a_n$ are positive integers and $k = \max\{a_1, \ldots, a_n\}$, $t = \min\{a_1,\ldots, a_n\}$, prove the inequality \[\left(\frac{a_1^2+a_2^2+\cdots+a_n^2}{a_1+a_2+\cdots+a_n}\right)^{\frac{kn}{t}} \geq a_1a_2\cdots a_n.\] When does equality hold?

Let $ABC$ be a triangle with side-lengths $a, b, c$, inscribed in a circle with radius $R$ and let $I$ be ir's incenter. Let $P_1, P_2$ and $P_3$ be the areas of the triangles $ABI, BCI$ and $CAI$, respectively. Prove that $$\frac{R^4}{P_1^2}+\frac{R^4}{P_2^2}+\frac{R^4}{P_3^2}\ge 16$$

The positive integers $a,b,c$ are less than $99$ and satisfy $a^2+b^2=c^2+99^2$. . Find the minimum and maximum value of $a+b+c$.