Let $f$ be a real-valued function such that \[f(x)+2f\left(\dfrac{2002}x\right)=3x\] for all $x>0$. Find $f(2)$. $\textbf{(A) }1000\qquad\textbf{(B) }2000\qquad\textbf{(C) }3000\qquad\textbf{(D) }4000\qquad\textbf{(E) }6000$

Prove that for every natural number $n$, and for every real number $x \neq \frac{k\pi}{2^t}$ ($t=0,1, \dots, n$; $k$ any integer) \[ \frac{1}{\sin{2x}}+\frac{1}{\sin{4x}}+\dots+\frac{1}{\sin{2^nx}}=\cot{x}-\cot{2^nx} \]

Let $\mathbb X$ be the set of all bijective functions from the set $S=\{1,2,\cdots, n\}$ to itself. For each $f\in \mathbb X,$ define \[T_f(j)=\left\{\begin{aligned} 1, \ \ \ & \text{if} \ \ f^{(12)}(j)=j,\\ 0, \ \ \ & \text{otherwise}\end{aligned}\right.\] Determine $\sum_{f\in\mathbb X}\sum_{j=1}^nT_{f}(j).$ (Here $f^{(k)}(x)=f(f^{(k-1)}(x))$ for all $k\geq 2.$)

The center of square $ABCD$ is $K$. The point $P$ is chosen such that $P \ne K$ and the angle $\angle APB$ is right . Prove that the line $PK$ bisects the angle between the lines $AP$ and $BP$.

On the "battleship" field (a square of $10\times 10$ cells), $10$ "ships" are placed in the following sequence: first one "ship" of size $1\times 4$, then two - of size $1\times 3$, three - of size $1\times 2$, and, finally, four - $1\times 1$. The rules do not allow "ships" to touch each other even with their tops. Can it happen that when part of the "ships" have already been displayed, there is nowhere to place the next one?

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

Let $ABCD$ be a quadrilateral whose diagonals are perpendicular, and let $S$ be the intersection of those diagonals. Let $K, L, M$ and $N$ be the reflections of $S$ on the sides $AB$, $BC$, $CD$ and $DA$ respectively. $BN$ cuts the circumcircle of $\vartriangle SKN$ at $E$ and $BM$ cuts the circumcircle of $\vartriangle SLM$ at $F$. Prove that the quadrilateral $EFLK$ is cyclic.

Prove that, for any positive integer $n$, $$1^n+2^n+3^n+4^n$$ is divisible by $5$ if and only if $n$ is not divisible by $4$.

A circle stands in a plane perpendicular to the ground and a point $A$ lies in this plane exterior to the circle and higher than its bottom. A particle starting from rest at $A$ slides without friction down an inclined straight line until it reaches the circle. Which straight line allows descent in the shortest time?

Points $A,M,B,C,D$ are given on a circle in this order such that $A$ and $B$ are equidistant from $M$. Lines $MD$ and $AC$ intersect at $E$ and lines $MC$ and $BD$ intersect at $F$. Prove that the quadrilateral $CDEF$ is inscridable in a circle.

Through point $ P $ inside triangle $ ABC $, straight lines were drawn, parallel to the sides, until they intersect with the sides. In the three resulting parallelograms, diagonals that do not contain point $ P $, are drawn. Points $ A_1 $, $ B_1 $ and $ C_1 $ are the intersection points of the lines containing these diagonals such that $A_1$ and $A$ are in different sides of line $BC$ and $B_1$ and $C_1$ are similar. Prove that if hexagon $ AC_1BA_1CB_1 $ is inscribed and convex, then point $ P $ is the orthocenter of triangle $ A_1B_1C_1 $.

Let $A_r,B_r, C_r$ be points on the circumference of a given circle $S$. From the triangle $A_rB_rC_r$, called $\Delta_r$, the triangle $\Delta_{r+1}$ is obtained by constructing the points $A_{r+1},B_{r+1}, C_{r+1} $on $S$ such that $A_{r+1}A_r$ is parallel to $B_rC_r$, $B_{r+1}B_r$ is parallel to $C_rA_r$, and $C_{r+1}C_r$ is parallel to $A_rB_r$. Each angle of $\Delta_1$ is an integer number of degrees and those integers are not multiples of $45$. Prove that at least two of the triangles $\Delta_1,\Delta_2, \ldots ,\Delta_{15}$ are congruent.

A [b][u]word[/u][/b] is formed by a number of letters of the alphabet. We show words with capital letters. A [b][u]sentence[/u][/b] is formed by a number of words. For example if $A=aa$ and $B=ab$ then the sentence $AB$ is equivalent to $aaab$. In this language, $A^n$ indicates $\underbrace{AA \cdots A}_{n}$. We have an equation when two sentences are equal. For example $XYX=YZ^2$ and it means that if we write the alphabetic letters forming the words of each sentence, we get two equivalent sequences of alphabetic letters. An equation is [b][u]simplified[/u][/b], if the words of the left and the right side of the sentences of the both sides of the equation are different. Note that every word contains one alphabetic letter at least. $\text{a})$We have a simplified equation in terms of $X$ and $Y$. Prove that both $X$ and $Y$ can be written in form of a power of a word like $Z$.($Z$ can contain only one alphabetic letter). $\text{b})$ Words $W_1,W_2,\cdots , W_n$ are the answers of a simplified equation. Prove that we can produce these $n$ words with fewer words. $\text{c})$ $n$ words $W_1,W_2,\cdots , W_n$ are the answers of a simplified system of equations. Define graph $G$ with vertices ${1,2 \cdots ,n}$ such that $i$ and $j$ are connected if in one of the equations, $W_i$ and $W_j$ be the two words appearing in the right side of each side of the equation.($\cdots W_i = \cdots W_j$). If we denote by $c$ the number of connected components of $G$, prove that these $n$ words can be produced with at most $c$ words. [i]Proposed by Mostafa Einollah Zadeh Samadi[/i]

$1000\times 1993 \times 0.1993 \times 10 = $ $\text{(A)}\ 1.993\times 10^3 \qquad \text{(B)}\ 1993.1993 \qquad \text{(C)}\ (199.3)^2 \qquad \text{(D)}\ 1,993,001.993 \qquad \text{(E)}\ (1993)^2$

Let a positive integer $n$.Consider square table $3*3$.One use $n$ colors to color all cell of table such that each cell is colored by exactly one color. Two colored table is same if we can receive them from other by a rotation through center of $3*3$ table How many way to color this square table satifies above conditions.

For how many ordered pairs $(x,y)$ of integers is it true that $0<x<y<10^{6}$ and that the arithmetic mean of $x$ and $y$ is exactly $2$ more than the geometric mean of $x$ and $y?$