$(CZS 3)$ Let $a$ and $b$ be two positive real numbers. If $x$ is a real solution of the equation $x^2 + px + q = 0$ with real coefficients $p$ and $q$ such that $|p| \le a, |q| \le b,$ prove that $|x| \le \frac{1}{2}(a +\sqrt{a^2 + 4b})$ Conversely, if $x$ satisfies the above inequality, prove that there exist real numbers $p$ and $q$ with $|p|\le a, |q|\le b$ such that $x$ is one of the roots of the equation $x^2+px+ q = 0.$

Show that $\frac{(x + y + z)^2}{3} \ge x\sqrt{yz} + y\sqrt{zx} + z\sqrt{xy}$ for all non-negative reals $x, y, z$.

Let $a,b,c$ be positive real numbers such that $a + b + c = 1$. Show that $$\left(\frac{1}{a} + \frac{1}{bc}\right)\left(\frac{1}{b} + \frac{1}{ca}\right)\left(\frac{1}{c} + \frac{1}{ab}\right) \geq 1728$$

Let $ A_0 \equal{} (a_1,\dots,a_n)$ be a finite sequence of real numbers. For each $ k\geq 0$, from the sequence $ A_k \equal{} (x_1,\dots,x_k)$ we construct a new sequence $ A_{k \plus{} 1}$ in the following way. 1. We choose a partition $ \{1,\dots,n\} \equal{} I\cup J$, where $ I$ and $ J$ are two disjoint sets, such that the expression \[ \left|\sum_{i\in I}x_i \minus{} \sum_{j\in J}x_j\right| \] attains the smallest value. (We allow $ I$ or $ J$ to be empty; in this case the corresponding sum is 0.) If there are several such partitions, one is chosen arbitrarily. 2. We set $ A_{k \plus{} 1} \equal{} (y_1,\dots,y_n)$ where $ y_i \equal{} x_i \plus{} 1$ if $ i\in I$, and $ y_i \equal{} x_i \minus{} 1$ if $ i\in J$. Prove that for some $ k$, the sequence $ A_k$ contains an element $ x$ such that $ |x|\geq\frac n2$. [i]Author: Omid Hatami, Iran[/i]

Let $x,y,z$ be positive real numbers such that $xyz=1$. Prove that $\frac{1}{(x+1)^2+y^2+1}$ $+$ $\frac{1}{(y+1)^2+z^2+1}$ $+$ $\frac{1}{(z+1)^2+x^2+1}$ $\leq$ ${\frac{1}{2}}$.

The positive real numbers $a, b, c$ satisfy the equation $a+b+c=1$. Prove the identity: $\sqrt{\frac{(a+bc)(b+ca)}{c+ab}}+\sqrt{\frac{(b+ca)(c+ab)}{a+bc}}+\sqrt{\frac{(c+ab)(a+bc)}{b+ca}} = 2$

If $x$, $y$ are positive reals such that $x + y = 2$ show that $x^3y^3(x^3+ y^3) \leq 2$.

Let $a,b,c\ge 0$ and $a+b+c=\sqrt2$. Show that \[\frac1{\sqrt{1+a^2}}+\frac1{\sqrt{1+b^2}}+\frac1{\sqrt{1+c^2}} \ge 2+\frac1{\sqrt3}\] [hide] In general if $a_1, a_2, \cdots , a_n \ge 0$ and $\sum_{i=1}^n a_i=\sqrt2$ we have \[\sum_{i=1}^n \frac1{\sqrt{1+a_i^2}} \ge (n-1)+\frac1{\sqrt3}\] [/hide]

Let $\theta_i \in \left ( 0,\frac{\pi}{4} \right ]$ for $i=1,2,3,4$. Prove that: $\tan \theta _1 \tan \theta _2 \tan \theta _3 \tan \theta _4 \le (\frac{\sin^8 \theta _1+\sin^8 \theta _2+\sin^8 \theta _3+\sin^8 \theta _4}{\cos^8 \theta _1+\cos^8 \theta _2+\cos^8 \theta _3+\cos^8 \theta _4})^\frac{1}{2}$ [hide=edit]@below, fixed now. There were some problems (weird characters) so aops couldn't send it.[/hide]

Show that \[ \frac{1 - s^a}{1 - s} \leq (1 + s)^{a-1}\] holds for every $1 \neq s > 0$ real and $0 < a \leq 1$ rational.

Find the greatest integer less than or equal to $\sum_{k=1}^{2^{1983}} k^{\frac{1}{1983} -1}.$

Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that $$\frac{a^2}{(b+c)^3}+\frac{b^2}{(c+a)^3}+\frac{c^2}{(a+b)^3}\geq \frac98$$

For real numbers $0 \leq a_1 \leq a_2 \leq ... \leq a_n$ and $0 \leq b_1 \leq b_2 \leq ... \leq b_n$ prove that \[ \left( \frac{a_1}{1 \cdot 2}+\frac{a_2}{2 \cdot 3}+...+\frac{a_n}{n(n+1)} \right) \times \left( \frac{b_1}{1 \cdot 2}+\frac{b_2}{2 \cdot 3}+...+\frac{b_n}{n(n+1)} \right) \leq \frac{a_1b_1}{1 \cdot 2}+\frac{a_2b_2}{2 \cdot 3}+...+\frac{a_nb_n}{n(n+1)}.\] [i]Proposed by A. Antropov[/i]

Given real numbers $x_1, x_2, \dots , x_n \ (n \geq 2)$, with $x_i \geq \frac 1n \ (i = 1, 2, \dots, n)$ and with $x_1^2+x_2^2+\cdots+x_n^2 = 1$ , find whether the product $P = x_1x_2x_3 \cdots x_n$ has a greatest and/or least value and if so, give these values.

Let $x, y,$ and $z$ be positive real numbers such that $xy + yz + zx = 27$. Prove that $x + y + z \ge \sqrt{3xyz}$. When does equality hold?

Let $x,y$ and $z$ be positive real numbers such that $xy+yz+xz=3xyz$. Prove that \[ x^2y+y^2z+z^2x \ge 2(x+y+z)-3 \] and determine when equality holds. [i]UK - David Monk[/i]

All sides of a convex polygon were decreased in such a way that they formed a new convex polygon. Is it possible that all diagonals were increased?

Let $a,b,c$ be positive real numbers, that satisfy $abc=1$. Prove the inequality: $$\frac{ab}{1+c}+\frac{bc}{1+a}+\frac{ca}{1+b} \geq \frac{27}{(a+b+c)(3+a+b+c)}$$

Let $f$ be an injective function from ${1,2,3,\ldots}$ in itself. Prove that for any $n$ we have: $\sum_{k=1}^{n} f(k)k^{-2} \geq \sum_{k=1}^{n} k^{-1}.$

Let $A$ and $B$ be given real numbers. Let the sum of real numbers $0\le x_1\le x_2\le\ldots \le x_n$ be $A$, and let the sum of real numbers $0\le y_1\le y_2\le \ldots\le y_n$ be $B$. Find the largest possible value of \[\sum_{i=1}^n (x_i-y_i)^2.\] [i]Proposed by Péter Csikvári, Budapest[/i]

Define the numbers $a_0, a_1, \ldots, a_n$ in the following way: \[ a_0 = \frac{1}{2}, \quad a_{k+1} = a_k + \frac{a^2_k}{n} \quad (n > 1, k = 0,1, \ldots, n-1). \] Prove that \[ 1 - \frac{1}{n} < a_n < 1.\]

For real numbers $a, b, c, d \in [0, 1]$, find the largest possible value of the following expression: $$a^2+b^2+c^2+d^2-ab-bc-cd-da$$ [i]Proposed by Mykhailo Shtandenko[/i]

Prove that for $\forall$ $z\in \mathbb{C}$ the following inequality is true: $|z|^2+2|z-1|\geq 1$, where $"="$ is reached when $z=1$.

Let $a, b$, and $c$ be the lengths of the sides of an arbitrary triangle, and let $\alpha,\beta$, and $\gamma$ be the radian measures of its corresponding angles. Prove that $$ \frac{\pi}{3}\le \frac{\alpha a +\beta b + \gamma c}{a+b+c} < \frac{\pi}{2}.$$ Suggest spatial analogues of this inequality.

Given real numbers $x_1,x_2,y_1,y_2,z_1,z_2$ satisfying $x_1>0,x_2>0,x_1y_1>z_1^2$, and $x_2y_2>z_2^2$, prove that: \[ {8\over(x_1+x_2)(y_1+y_2)-(z_1+z_2)^2}\le{1\over x_1y_1-z_1^2}+{1\over x_2y_2-z_2^2}. \] Give necessary and sufficient conditions for equality.