Solve in real numbers the equation $$\left(x^2-3x-2\right)^2-3\left(x^2-3x-2\right)-2-x=0.$$

Let $ a_0$, $ a_1$, $ a_2$, $ \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $ \gcd (a_i, a_{i \plus{} 1}) > a_{i \minus{} 1}$. Prove that $ a_n\ge 2^n$ for all $ n\ge 0$. [i]Proposed by Morteza Saghafian, Iran[/i]

Let $ABC$ be an acute angled triangle and $\Gamma$ its circumcircle. Led $D$ be a point on segment $BC$, different from $B$ and $C$, and let $M$ be the midpoint of $AD$. The line perpendicular to $AB$ that passes through $D$ intersects $AB$ in $E$ and $\Gamma$ in $F$, with point $D$ between $E$ and $F$. Lines $FC$ and $EM$ intersect at point $X$. If $\angle DAE = \angle AFE$, show that line $AX$ is tangent to $\Gamma$.

Let $a,b,c,x,y,z$ be real numbers such that $x+y+z=0$, $a+b+c\geq 0$, $ab+bc+ca \ge 0$. Prove that $$ ax^2+by^2+cz^2\ge 0 $$

In a certain language there are only two letters, $A$ and $B$. The words of this language obey the following rules: (i) The only word of length $1$ is $A$; (ii) A sequence of letters $X_1X_2...X_{n+1}$, where $X_i\in \{A,B\}$ for each $i$, forms a word of length $n+1$ if and only if it contains at least one letter $A$ and is not of the form $WA$ for a word $W$ of length $n$. Show that the number of words consisting of $1998 A$’s and $1998 B$’s and not beginning with $AA$ equals $\binom{3995}{1997}-1$

Let $ a_1\geq \cdots \geq a_n \geq a_{n \plus{} 1} \equal{} 0$ be real numbers. Show that \[ \sqrt {\sum_{k \equal{} 1}^n a_k} \leq \sum_{k \equal{} 1}^n \sqrt k (\sqrt {a_k} \minus{} \sqrt {a_{k \plus{} 1}}). \] [i]Proposed by Romania[/i]

Given a positive integer $ n$, find all integers $ (a_{1},a_{2},\cdots,a_{n})$ satisfying the following conditions: $ (1): a_{1}\plus{}a_{2}\plus{}\cdots\plus{}a_{n}\ge n^2;$ $ (2): a_{1}^2\plus{}a_{2}^2\plus{}\cdots\plus{}a_{n}^2\le n^3\plus{}1.$

For a set \(S\) of positive integers and a positive integer \(n\), consider the game of [i]\((n,S)\)-nim[/i], which is as follows. A pile starts with \(n\) watermelons. Two players, Deric and Erek, alternate turns eating watermelons from the pile, with Deric going first. On any turn, the number of watermelons eaten must be an element of \(S\). The last player to move wins. Let \(f(S)\) denote the set of positive integers \(n\) for which Deric has a winning strategy in \((n,S)\)-nim. Let \(T\) be a set of positive integers. Must the sequence \[T, \; f(T), \; f(f(T)), \;\ldots\] be eventually constant? [i]Proposed by Brandon Wang and Edward Wan[/i]

Let $a,b,c$ be positive reals. Prove that $$a^ab^bc^c \geq a^bb^cc^a$$

Let $f(x,y)$ be a polynomial with real coefficients in the real variables $x$ and $y$ defined over the entire $xy$-plane. What are the possibilities for the range of $f(x,y)?$

The train consists of six cars. On average, each carriage carries $18$ passengers. After one car was uncoupled, the average number of passengers in the remaining cars was reduced to $15$. How many passengers were in the uncoupled car?

A square $ ABCD$ of side length $ 2$ is given on a plane. The segment $ AB$ is moved continuously towards $ CD$ until $ A$ and $ C$ coincide with $ C$ and $ D$, respectively. Let $ S$ be the area of the region formed by the segment $ AB$ while moving. Prove that $ AB$ can be moved in such a way that $ S <\frac{5\pi}{6}$.

Let $ABC$ be an acute-angled scalene triangle. Inside the triangle, distinct points $A_1, B_1, C_1$ are chosen such that \[ \frac{AB_1}{CB_1} = \frac{AB}{CB} \quad \text{and} \quad \frac{AC_1}{BC_1} = \frac{AC}{BC}. \] Let $A_2, B_2, C_2$ be the reflections of $A_1, B_1, C_1$ across lines $BC, AC, AB$, respectively. These points satisfy the following conditions: [list] [*] The four points $A, A_2, B, C_2$ are concyclic. [*] The four points $A, A_2, B_2, C$ are concyclic. [*] The four points $B, B_2, C, C_2$ are concyclic. [*] The three points $A_2, B_2, C_2$ do not lie on the circumcircle of $\triangle ABC$. [/list] Prove that triangles $A_1B_1C_1$ and $A_2B_2C_2$ are similar.

Segment $AD$ is the diameter of the circumscribed circle of an acute-angled triangle $ABC$. Through the intersection of the altitudes of this triangle, a straight line was drawn parallel to the side $BC$, which intersects sides $AB$ and $AC$ at points $E$ and $F$, respectively. Prove that the perimeter of the triangle $DEF$ is two times larger than the side $BC$.

Let $M$ be the set of the vertices of a regular hexagon, our Olympiad symbol. How many chains $\emptyset \subset A \subset B \subset C \subset D \subset M$ of six different set, beginning with the empty set and ending with the $M$, are there?

Given a real number $ c > 0$, a sequence $ (x_n)$ of real numbers is defined by $ x_{n \plus{} 1} \equal{} \sqrt {c \minus{} \sqrt {c \plus{} x_n}}$ for $ n \ge 0$. Find all values of $ c$ such that for each initial value $ x_0$ in $ (0, c)$, the sequence $ (x_n)$ is defined for all $ n$ and has a finite limit $ \lim x_n$ when $ n\to \plus{} \infty$.