If $x$, $y$, $z$ are positive numbers satisfying \[x+\frac{y}{z}=y+\frac{z}{x}=z+\frac{x}{y}=2.\] Find all the possible values of $x+y+z$.

Let $x, y$ be positive reals such that $x \ne y$. Find the minimum possible value of $(x + y)^2 + \frac{54}{xy(x-y)^2}$ .

Acute-angled $~$ $\triangle{ABC}$ $~$ is given. A line $~$ $l$ $~$ parallel to side $~$ $AB$ $~$ passing through vertex $~$ $C$ $~$ is drawn. Let the angle bisectors of $~$ $\angle{BAC}$ $~$ and $~$ $\angle{ABC}$ $~$ intersect the sides $~$ $BC$ and $~$ $AC$ at points $~$ $D$ $~$ and $~$ $F$, and line $~$ $l$ $~$ at points $~$ $E$ $~$ and $~$ $G$ $~$ respectively. Prove that if $~$ $\overline{DE}=\overline{GF}$ $~$ then $~$ $\overline{AC}=\overline{BC}\, .$

Let $0 \leq \alpha , \beta , \gamma \leq \frac{\pi}{2}$, such that $\cos ^{2} \alpha + \cos ^{2} \beta + \cos ^{2} \gamma = 1$. Prove that $2 \leq (1 + \cos ^{2} \alpha ) ^{2} \sin^{4} \alpha + (1 + \cos ^{2} \beta ) ^{2} \sin ^{4} \beta + (1 + \cos ^{2} \gamma ) ^{2} \sin ^{4} \gamma \leq (1 + \cos ^{2} \alpha )(1 + \cos ^{2} \beta)(1 + \cos ^{2} \gamma ).$

Suppose that $ ABC$ is an acute triangle such that $ \tan A$, $ \tan B$, $ \tan C$ are the three roots of the equation $ x^3 \plus{} px^2 \plus{} qx \plus{} p \equal{} 0$, where $ q\neq 1$. Show that $ p \le \minus{} 3\sqrt 3$ and $ q > 1$.

Professor Oak is feeding his $100$ Pokémon. Each Pokémon has a bowl whose capacity is a positive real number of kilograms. These capacities are known to Professor Oak. The total capacity of all the bowls is $100$ kilograms. Professor Oak distributes $100$ kilograms of food in such a way that each Pokémon receives a non-negative integer number of kilograms of food (which may be larger than the capacity of the bowl). The [i]dissatisfaction level[/i] of a Pokémon who received $N$ kilograms of food and whose bowl has a capacity of $C$ kilograms is equal to $\lvert N-C\rvert$. Find the smallest real number $D$ such that, regardless of the capacities of the bowls, Professor Oak can distribute food in a way that the sum of the dissatisfaction levels over all the $100$ Pokémon is at most $D$. [i]Oleksii Masalitin, Ukraine[/i]

Let $P(x)$ be a monic polynomial of degree $n$ with nonnegative coefficients and the free term equal to $1$. Prove that if all the roots of $P(x)$ are real, then $P(x) \geq (x+1)^{n}$ holds for every $x \geq 0$.

Find the number of pairs of real numbers $(x,y)$ such that $x^4+y^4=4xy-2$.

$\triangle ABC$ has semiperimeter $s$ and area $F$ . A square $P QRS$ with side length $x$ is inscribed in $ABC$ with $P$ and $Q$ on $BC$, $R$ on $AC$, and $S$ on $AB$. Similarly, $y$ and $z$ are the sides of squares two vertices of which lie on $AC$ and $AB$, respectively. Prove that \[\frac 1x +\frac 1y + \frac 1z \le \frac{s(2+\sqrt3)}{2F}\]

Let $x,y,z$ be real numbers such that $0<x,y,z<1$. Find the minimum value of: $$\frac {xyz(x+y+z)+(xy+yz+zx)(1-xyz)}{xyz\sqrt {1-xyz}}$$

Points $B_1,C_1$ are on $AC$ and $AB$ and $B_1C_1 \parallel BC$. Circumcircle of $ABB_1$ intersect $CC_1$ at $L$. Circumcircle $CLB_1$ is tangent to $AL$. Prove $AL \leq \frac{AC+AC_1}{2}$

Let \( n > 2 \) be an integer, \( k > 1 \) a real number, and \( x_1, x_2, \ldots, x_n \) be positive real numbers such that \( x_1 \cdot x_2 \cdots x_n = 1 \). Prove that: \[ \frac{1 + x_1^k}{1 + x_2} + \frac{1 + x_2^k}{1 + x_3} + \cdots + \frac{1 + x_n^k}{1 + x_1} \geq n. \] When does equality hold?

Solve the system of equations: \[ x+y+z=a \] \[ x^2+y^2+z^2=b^2 \] \[ xy=z^2 \] where $a$ and $b$ are constants. Give the conditions that $a$ and $b$ must satisfy so that $x,y,z$ are distinct positive numbers.

Let $a,b,c$ be integers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=3$ prove that $abc$ is a perfect cube!

If $a,b,c\in (0,1)$ satisfy $a+b+c=2$ , prove that $\frac{abc}{(1-a)(1-b)(1-c)}\ge 8$