a) Each of the numbers $x_1,x_2,...,x_n$ can be $1, 0$, or $-1$. What is the minimal possible value of the sum of all products of couples of those numbers. b) Each absolute value of the numbers $x_1,x_2,...,x_n$ doesn't exceed $1$. What is the minimal possible value of the sum of all products of couples of those numbers.

Let $a, b$ and $c$ be positive real numbers such that $a+b+c = abc$. Prove that at least one of $a, b$ or $c$ is greater than $\frac{17}{10}$ .

Consider the following inequality for real numbers $x,y,z$: $|x-y|+|y-z|+|z-x| \le a \sqrt{x^2 +y^2 +z^2}$ . (a) Prove that the inequality is valid for $a = 2\sqrt2$ (b) Assuming that $x,y,z$ are nonnegative, show that the inequality is also valid for $a = 2$.

Prove theat for an arbitrary triangle holds the inequality $$a \cos A+ b \cos B + c \cos C \le p ,$$ where $a, b, c$ are the sides of the triangle, $A, B, C$ are the angles, $p$ is the semiperimeter.

Let $ x_1,x_2,\ldots,x_n$ be real numbers greater than 1. Show that \[ \frac{x_1x_2}{x_3}\plus{}\frac{x_2x_3}{x_4}\plus{}\cdots\plus{}\frac{x_nx_1}{x_2}\ge4n\] and determine when the equality holds.

A positive integer $N$ has $20$ digits when written in base $9$ and $13$ digits when written in base $27$. How many digits does $N$ have when written in base $3$? [i]Proposed by Aaron Lin[/i]

Find all values of the parameter $a$ for which there exist exactly two integer values of $x$ that satisfy the inequality $$x^2+5\sqrt2 x+a<0.$$

If $a$, $b$, $c > 0$; $ab + bc + ca = 3$ then: $$4\left(\tan^{-1} 2\right)\left(\tan^{-1}\left(\sqrt[3]{abc}\right)\right) \leq \pi \tan^{-1}\left(1 + \sqrt[3]{abc}\right)$$

Let $a, b, c$ be non-negative reals with $a + b + c \le 2$. prove that $$\sqrt{b^2+ac} + \sqrt{a^2+bc} + \sqrt{c^2+ab} \le 3$$

Let $a,b,c\ge0$ and $a+b+c\ge3.$ Prove that $a^4+b^3+c^2\ge a^3+b^2+c.$

Let $\ a,x,y,z>0$. Prove that: $\frac{a+y}{a+z}x+\frac{a+z}{a+x}y+\frac{a+x}{a+y}z\geq{x+y+z}\geq\frac{a+z}{a+x}x+\frac{a+x}{a+y}y+\frac{a+y}{a+z}z$

Prove that for each $m\geq1$: \[\sum_{|k|<\sqrt m}\binom{2m}{m+k}\geq 2^{2m-1}\] [hide="Hint"]Maybe probabilistic method works[/hide]

Let $x_1,x_2,\ldots,x_{2001}$ be positive numbers such that $$x_i^2\ge x_1^2+\frac{x_2^2}{2^3}+\frac{x_3^2}{3^3}+\ldots+\frac{x_{i-1}^2}{(i-1)^3}\enspace\text{for }2\le i\le2001.$$Prove that $\sum_{i=2}^{2001}\frac{x_i}{x_1+x_2+\ldots+x_{i-1}}>1.999$.

Let \begin{eqnarray*} f(x) & = & a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 \ \ \mbox{and} \\ g(x) & = & c_{n+1} x^{n+1} + c_n x^n + \cdots + c_0 \end{eqnarray*} be non-zero polynomials with real coefficients such that $g(x) = (x+r)f(x)$ for some real number $r$. If $a = \max(|a_n|, \ldots, |a_0|)$ and $c = \max(|c_{n+1}|, \ldots, |c_0|)$, prove that $\frac{a}{c} \leq n+1$.

Prove that for all positive real numbers $a, b, c$ with $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} =1$ the following inequality holds: $$3(ab+bc+ca)+\frac{9}{a+b+c} \le \frac{9abc}{a+b+c} + 2(a^2+b^2+c^2)+1$$

Positive real numbers $a,b,c$ satisfy $$2a^3b+2b^3c+2c^3a=a^2b^2+b^2c^2+c^2a^2.$$ Prove that $$2ab(a-b)^2+2bc(b-c)^2+2ca(c-a)^2 \geq(ab+bc+ca)^2.$$

For each pair $(\alpha, \beta)$ of non-negative reals with $\alpha+\beta \geq 2$, determine all functions $f:\mathbb{R} \rightarrow \mathbb{R}$, such that $$f(x)f(y) \leq f(xy)+\alpha x+\beta y$$ for all reals $x, y$.

Prove that for any positive reals $ a,b,c,d $ with $ a+b+c+d = 4 $, we have $$ \sum\limits_{cyc}{\frac{3a^3}{a^2+ab+b^2}}+\sum\limits_{cyc}{\frac{2ab}{a+b}} \ge 8 $$

Let $f$ be a function that is defined for all positive integers and whose values are positive integers. For $f$ it also holds that $f (n + 1)> f (n)$ and $f (f (n)) = 3n$, for each positive integer $n$. Calculate $f (2019)$.

In $\mathbb{R}^3$, for a rectangular box $\Delta$, let $10\Delta$ be the box with the same center as $\Delta$ but dilated by $10$. For example, if $\Delta$ is an $1 \times 1 \times 10$ box (hence with Lebesgue measure $10$), then $10\Delta$ is the $10 \times 10 \times 100$ box with the same center and orientation as $\Delta$. \medskip If two rectangular boxes $\Delta_1$ and $\Delta_2$ satisfy $\Delta_1 \subset 10\Delta_2$ and $\Delta_2 \subset 10 \Delta_1$, we say that they are [i]almost identical[/i]. Find the largest real number $a$ such that the following holds for some $C=C(a)>0$: For every positive integer $N$ and every collection $S$ of $1 \times 1 \times N$ boxes in $\mathbb{R}^3$, assuming that (i) $\vert S\vert=N$, (ii) every pair of boxes $(\Delta_1,\Delta_2)$ taken from $S$ are not almost identical, and (iii) the long edge of each box in $S$ forms an angle $\frac{\pi}{4}$ against the $xy$-plane. Then the volume \[\left\vert \bigcup_{\Delta \in S} \Delta\right\vert \ge CN^a.\]

Let $P(x)$ be a polynomial of degree $n$ with real coefficients and let $a\geq 3$. Prove that \[\max_{0\leq j \leq n+1}\left | a^j-P(j) \right |\geq 1\]

Let $n \in \mathbb{N} , n \ge 2$ and $ a_0,a_1,a_2,\cdots,a_n \in \mathbb{C} ; a_n \not = 0 $. Then: [b][size=100][i]P.[/i][/size][/b] $|a_nz^n + a_{n-1}z^z{n-1} + \cdots + a_1z + a_0 | \le |a_n+a_0|$ for any $z \in \mathbb{C}, |z|=1$ [b][size=100][i]Q[/i][/size][/b]. $a_1=a_2=\cdots=a_{n-1}=0$ and $a_0/a_n \in [0,\infty)$ Prove that $ P \Longleftrightarrow Q$

If $x_1$ and $x_2$ are the solutions of the equation $x^2-(m+3)x+m+2=0$ Find all real values of $m$ such that the following inequations are valid $\frac {1}{x_1}+\frac {1}{x_2}>\frac{1}{2}$ and $x_1^2+x_2^2<5$

Let $\mathbb{Z}^{+}$ be the set of positive integers. [b]a)[/b] Prove that there is only one function $f:\mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}$, strictly increasing, such that $f(f(n))=2n+1$ for every $n\in \mathbb{Z}^{+}$. [b]b)[/b] For the function in [b]a[/b]. Prove that for every $n\in \mathbb{Z}^{+}$ $\frac{4n+1}{3}\leq f(n)\leq \frac{3n+1}{2}$ [b]c) [/b] Prove that in each inequality side of [b]b[/b] the equality can reach by infinite positive integers $n$.

Let $a$ and $b$ be two positive reals such that the following inequality \[ ax^3 + by^2 \geq xy - 1 \] is satisfied for any positive reals $x, y \geq 1$. Determine the smallest possible value of $a^2 + b$. [i]Proposed by Fajar Yuliawan[/i]