Each deputy of the Academic Duma quarreled with exactly three other deputies. The President ordered the Speaker to divide the deputies into n factions so that agreement reigned within one faction. For what smallest $n$ is this always possible? (This means that there is such $n$ that deputies could always be divided into $n$ factions, but not always into $(n- 1)$ factions.)

Find all functions $f:R\rightarrow R$ such that \[ f(x^3)+f(y^3)=(x+y)(f(x^2)+f(y^2)-f(xy)) \] for all $x,y\in R$.

A car has a 4-digit integer price, which is written digitally. (so in digital numbers, like on your watch probably) While the salesmen isn't watching, the buyer turns the price upside down and gets the car for 1626 less. How much did the car initially cost?

In a square $ABCD$, $E$ is the midpoint of side $BC$. Line $AE$ intersects line $DC$ at $F$ and diagonal $BD$ at $G$. If the area $(EFC) = 8$, determine the area $(GBE)$.

Let $a, b, c$ be positive integers whose greatest common divisor is $1$. Determine whether there always exists a positive integer $n$ such that, for every positive integer $k$, the number $2^n$ is not a divisor of $a^k+b^k+c^k$.

Inside triangle \( ABC \), a point \( D \) is chosen such that \( \angle ADB = \angle ADC \). The rays \( BD \) and \( CD \) intersect the circumcircle of triangle \( ABC \) at points \( E \) and \( F \), respectively. On segment \( EF \), points \( K \) and \( L \) are chosen such that \linebreak \( \angle AKD = 180^\circ - \angle ACB \) and \( \angle ALD = 180^\circ - \angle ABC \), with segments \( EL \) and \( FK \) \linebreak not intersecting line \( AD \). Prove that \( AK = AL \). [i]Proposed by Matthew Kurskyi[/i]

The set of all real numbers $ x$ for which \[ \log_{2004}(\log_{2003}(\log_{2002}(\log_{2001}{x}))) \]is defined is $ \{x|x > c\}$. What is the value of $ c$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2001^{2002} \qquad \textbf{(C)}\ 2002^{2003} \qquad \textbf{(D)}\ 2003^{2004} \qquad \textbf{(E)}\ 2001^{2002^{2003}}$

In triangle $ABC$, $R$ is the midpoint of $BC$ and $CS = 3SA$. If $x$ is the area of $CRS$, $y$ is the area of $RBT$, $z$ is the area of $ATS$, and $y^2 = xz$, then what is the value of $\frac{AT}{TB}$? Express your answer in the form $\frac{a+b\sqrt{c}}{d}$ , where $a,b,c,d$ are integers, $d$ is positive and as small as possible, and $c$ is squarefree. [img]https://cdn.artofproblemsolving.com/attachments/f/d/65b443628329610ff41d30b95e5ebd0c914f20.jpg[/img]

You have two blackboards $A$ and $B$. You have to write on them some of the integers greater than or equal to $2$ and less than or equal to $20$ in such a way that each number on blackboard $A$ is co-prime with each number on blackboard $B.$ Determine the maximum possible value of multiplying the number of numbers written in $A$ by the number of numbers written in $B$.

We mean a traingle in $\mathbb Q^{n}$, 3 points that are not collinear in $\mathbb Q^{n}$ a) Suppose that $ABC$ is triangle in $\mathbb Q^{n}$. Prove that there is a triangle $A'B'C'$ in $\mathbb Q^{5}$ that $\angle B'A'C'=\angle BAC$. b) Find a natural $m$ that for each traingle that can be embedded in $\mathbb Q^{n}$ it can be embedded in $\mathbb Q^{m}$. c) Find a triangle that can be embedded in $\mathbb Q^{n}$ and no triangle similar to it can be embedded in $\mathbb Q^{3}$. d) Find a natural $m'$ that for each traingle that can be embedded in $\mathbb Q^{n}$ then there is a triangle similar to it, that can be embedded in $\mathbb Q^{m}$. You must prove the problem for $m=9$ and $m'=6$ to get complete mark. (Better results leads to additional mark.)

Show that for any real numbers $ a,b, $ there exists $ c\in [-2,1] $ such that $ \big| c^3+ac+b\big| \ge 1. $

Three circles $O_1, \ O_2, \ O_3$ with radiii $r_1, \ r_2, \ r_3$ respectively are tangent extarnally in pairs. Let r be the radius of the inscrined circle of triangle $O_1O_2O_3$. Prove that $$ r=\sqrt{\dfrac{r_1r_2r_3}{r_1+r_2+r_3}}.$$

Define the sequences $(a_n),(b_n)$ by \begin{align*} & a_n, b_n > 0, \forall n\in\mathbb{N_+} \\ & a_{n+1} = a_n - \frac{1}{1+\sum_{i=1}^n\frac{1}{a_i}} \\ & b_{n+1} = b_n + \frac{1}{1+\sum_{i=1}^n\frac{1}{b_i}} \end{align*} 1) If $a_{100}b_{100} = a_{101}b_{101}$, find the value of $a_1-b_1$; 2) If $a_{100} = b_{99}$, determine which is larger between $a_{100}+b_{100}$ and $a_{101}+b_{101}$.

$ K$ takes $ 30$ minutes less time than $ M$ to travel a distance of $ 30$ miles. $ K$ travels $ \frac {1}{3}$ mile per hour faster than $ M$. If $ x$ is $ K$'s rate of speed in miles per hours, then $ K$'s time for the distance is: $ \textbf{(A)}\ \dfrac{x \plus{} \frac {1}{3}}{30} \qquad\textbf{(B)}\ \dfrac{x \minus{} \frac {1}{3}}{30} \qquad\textbf{(C)}\ \dfrac{30}{x \plus{} \frac {1}{3}} \qquad\textbf{(D)}\ \frac {30}{x} \qquad\textbf{(E)}\ \frac {x}{30}$

Let $S = \{\frac{\pi^n}{1992^m} | m,n \in \mathbb Z \}.$ Show that every real number $x \geq 0$ is an accumulation point of $S.$

Let $ k$ and $ k'$ two concentric circles centered at $ O$, with $ k'$ being larger than $ k$. A line through $ O$ intersects $ k$ at $ A$ and $ k'$ at $ B$ such that $ O$ seperates $ A$ and $ B$. Another line through $ O$ intersects $ k$ at $ E$ and $ k'$ at $ F$ such that $ E$ separates $ O$ and $ F$. Show that the circumcircle of $ \triangle{OAE}$ and the circles with diametres $ AB$ and $ EF$ have a common point.

Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.

Denote by $T(n)$ the sum of the digits of the decimal representation of a positive integer $n$. a) Find an integer $N$, for which $T(k \cdot N)$ is even for all $k, 1 \le k \le 1992, $ but $T(1993 \cdot N)$ is odd. b) Show that no positive integer $N$ exists such that $T(k \cdot N)$ is even for all positive integers $k$.

Let $ABCD$ be an isosceles trapezoid with $AD=BC=15$ such that the distance between its bases $AB$ and $CD$ is $7$. Suppose further that the circles with diameters $\overline{AD}$ and $\overline{BC}$ are tangent to each other. What is the area of the trapezoid?

Barry wrote 6 different numbers, one on each side of 3 cards, and laid the cards on a table, as shown. The sums of the two numbers on each of the three cards are equal. The three numbers on the hidden sides are prime numbers. What is the average of the hidden prime numbers? [asy]path card=((0,0)--(0,3)--(2,3)--(2,0)--cycle); draw(card, linewidth(1)); draw(shift(2.5,0)*card, linewidth(1)); draw(shift(5,0)*card, linewidth(1)); label("$44$", (1,1.5)); label("$59$", shift(2.5,0)*(1,1.5)); label("$38$", shift(5,0)*(1,1.5));[/asy] $ \textbf{(A)}\ 13 \qquad \textbf{(B)}\ 14 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 17$

For any positive numbers $a, b, c$ prove the inequalities \[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge \frac{2}{a+b}+\frac{2}{b+c}+\frac{2}{c+a}\ge \frac{9}{a+b+c}.\]

In a square $ABCD$, $K$ is a point on the side $BC$ and the bisector of $\angle KAD$ cuts the side $CD$ at the point $M$. Prove that the length of segment $AK$ is equal to the sum of the lengths of segments $DM$ and $BK$. (Folklore)

Let $n$ be an given positive integer, the set $S=\{1,2,\cdots,n\}$.For any nonempty set $A$ and $B$, find the minimum of $|A\Delta S|+|B\Delta S|+|C\Delta S|,$ where $C=\{a+b|a\in A,b\in B\}, X\Delta Y=X\cup Y-X\cap Y.$

Given a sequence $\{a_n\}$ of real numbers such that $|a_{k+m} - a_k - a_m| \leq 1$ for all positive integers $k$ and $m$, prove that, for all positive integers $p$ and $q$, \[|\frac{a_p}{p} - \frac{a_q}{q}| < \frac{1}{p} + \frac{1}{q}.\]

Considering all numbers of the form $n = \lfloor \frac{k^3}{2012} \rfloor$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$, and $k$ ranges from $1$ to $2012$, how many of these $n$’s are distinct?